1. **Problem:** Find the sum of $(x + 5) + (x^2 - 3x + 7)$. **Step 1:** Write the expression: $$ (x + 5) + (x^2 - 3x + 7) $$ **Step 2:** Combine like terms: $$ x^2 + (x - 3x) + (5 + 7) $$ **Step 3:** Simplify: $$ x^2 - 2x + 12 $$ 2. **Problem:** Find the difference $(7m - 8n^2 + 3n) - (-2n^2 + 4m - 3n)$. **Step 1:** Write the expression: $$ (7m - 8n^2 + 3n) - (-2n^2 + 4m - 3n) $$ **Step 2:** Distribute the minus sign: $$ 7m - 8n^2 + 3n + 2n^2 - 4m + 3n $$ **Step 3:** Combine like terms: $$ (7m - 4m) + (-8n^2 + 2n^2) + (3n + 3n) $$ **Step 4:** Simplify: $$ 3m - 6n^2 + 6n $$ 3. **Problem:** Find the total area to be carpeted for two rectangles: one $x$ by $x+3$, the other $(x-2)$ by $(x+5)$. **Step 1:** Area of first rectangle: $$ x(x + 3) = x^2 + 3x $$ **Step 2:** Area of second rectangle: $$ (x - 2)(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10 $$ **Step 3:** Total area: $$ (x^2 + 3x) + (x^2 + 3x - 10) = 2x^2 + 6x - 10 $$ **Answer:** Choice B: $2x^2 + 6x - 10$ 4. **Problem:** Find the product $a(a^2 + 2a - 10)$. **Step 1:** Distribute $a$: $$ a \cdot a^2 + a \cdot 2a + a \cdot (-10) = a^3 + 2a^2 - 10a $$ 5. **Problem:** Find the product $(2n - 5)(3a + 5)$. **Step 1:** Use distributive property: $$ 2n \cdot 3a + 2n \cdot 5 - 5 \cdot 3a - 5 \cdot 5 = 6an + 10n - 15a - 25 $$ 6. **Problem:** Find the product $(x - 3)(x^2 + 5x - 6)$. **Step 1:** Distribute $x$: $$ x \cdot x^2 + x \cdot 5x + x \cdot (-6) = x^3 + 5x^2 - 6x $$ **Step 2:** Distribute $-3$: $$ -3 \cdot x^2 - 3 \cdot 5x - 3 \cdot (-6) = -3x^2 - 15x + 18 $$ **Step 3:** Add results: $$ x^3 + 5x^2 - 6x - 3x^2 - 15x + 18 = x^3 + 2x^2 - 21x + 18 $$ 7. **Problem:** Find the product $(x + 3)^2$. **Step 1:** Use formula $(a + b)^2 = a^2 + 2ab + b^2$: $$ x^2 + 2 \cdot x \cdot 3 + 3^2 = x^2 + 6x + 9 $$ 8. **Problem:** Find the product $(2b - 5)(2b + 5)$. **Step 1:** Use difference of squares formula: $$ (2b)^2 - 5^2 = 4b^2 - 25 $$ 9a. **Problem:** Explain the expression $4000(1 + 0.05)^2$ for a 2-year CD at 5% interest. **Step 1:** $4000$ is the principal amount invested. **Step 2:** $0.05$ is the annual interest rate (5%). **Step 3:** $(1 + 0.05)^2$ accounts for compound interest over 2 years. 9b. **Problem:** Find the amount after 2 years. **Step 1:** Calculate: $$ 4000(1.05)^2 = 4000 \times 1.1025 = 4410 $$ 9c. **Problem:** Find amount after 4 years for $10,000$ at 6.25%. **Step 1:** Calculate: $$ 10000(1 + 0.0625)^4 = 10000(1.0625)^4 $$ **Step 2:** Compute $(1.0625)^4 \approx 1.28368$ **Step 3:** Multiply: $$ 10000 \times 1.28368 = 12836.8 $$ 10. **Problem:** Find the width of a rectangle with area $2x^2 - x - 15$ and length $2x + 5$. **Step 1:** Let width be $w$. Then: $$ (2x + 5)w = 2x^2 - x - 15 $$ **Step 2:** Solve for $w$: $$ w = \frac{2x^2 - x - 15}{2x + 5} $$ **Step 3:** Factor numerator: $$ 2x^2 - x - 15 = (2x + 5)(x - 3) $$ **Step 4:** Cancel common factor: $$ w = \frac{\cancel{(2x + 5)}(x - 3)}{\cancel{2x + 5}} = x - 3 $$ **Answer:** Choice H: $x - 3$ 11. **Problem:** Solve $5(t^2 - 3t + 2) = t(5t - 2)$. **Step 1:** Expand both sides: $$ 5t^2 - 15t + 10 = 5t^2 - 2t $$ **Step 2:** Subtract $5t^2$ from both sides: $$ \cancel{5t^2} - 15t + 10 = \cancel{5t^2} - 2t \implies -15t + 10 = -2t $$ **Step 3:** Add $15t$ to both sides: $$ 10 = 13t $$ **Step 4:** Solve for $t$: $$ t = \frac{10}{13} $$ 12. **Problem:** Solve $3x(x + 2) = 3(x^2 - 2)$. **Step 1:** Expand both sides: $$ 3x^2 + 6x = 3x^2 - 6 $$ **Step 2:** Subtract $3x^2$ from both sides: $$ \cancel{3x^2} + 6x = \cancel{3x^2} - 6 \implies 6x = -6 $$ **Step 3:** Solve for $x$: $$ x = -1 $$ 13. **Problem:** Factor $5xy - 10x$. **Step 1:** Factor out common factor $5x$: $$ 5x(y - 2) $$ 14. **Problem:** Factor $7ab + 14ab^2 + 21a^2b$. **Step 1:** Find common factor $7ab$: $$ 7ab(1 + 2b + 3a) $$ 15. **Problem:** Factor $4x^2 + 8x + x + 2$. **Step 1:** Group terms: $$ (4x^2 + 8x) + (x + 2) $$ **Step 2:** Factor each group: $$ 4x(x + 2) + 1(x + 2) $$ **Step 3:** Factor out common binomial: $$ (x + 2)(4x + 1) $$ 16. **Problem:** Factor $10a^2 - 50a - a + 5$. **Step 1:** Group terms: $$ (10a^2 - 50a) - (a - 5) $$ **Step 2:** Factor each group: $$ 10a(a - 5) - 1(a - 5) $$ **Step 3:** Factor out common binomial: $$ (a - 5)(10a - 1) $$ 17. **Problem:** Solve $y(y - 14) = 0$. **Step 1:** Set each factor equal to zero: $$ y = 0 \quad \text{or} \quad y - 14 = 0 $$ **Step 2:** Solve second equation: $$ y = 14 $$ **Answer:** $y = 0$ or $y = 14$ 18. **Problem:** Solve $3x(x + 6) = 0$. **Step 1:** Set each factor equal to zero: $$ 3x = 0 \quad \text{or} \quad x + 6 = 0 $$ **Step 2:** Solve: $$ x = 0 \quad \text{or} \quad x = -6 $$ 19. **Problem:** Solve $a^2 = 12a$. **Step 1:** Rearrange: $$ a^2 - 12a = 0 $$ **Step 2:** Factor: $$ a(a - 12) = 0 $$ **Step 3:** Set each factor to zero: $$ a = 0 \quad \text{or} \quad a - 12 = 0 $$ **Step 4:** Solve second: $$ a = 12 $$ 20. **Problem:** Find length if area is $x^2 - 100$ and width is $x - 10$. **Step 1:** Let length be $L$. Then: $$ L(x - 10) = x^2 - 100 $$ **Step 2:** Factor $x^2 - 100$: $$ (x - 10)(x + 10) $$ **Step 3:** Cancel common factor: $$ L = \frac{(x - 10)(x + 10)}{x - 10} = x + 10 $$ **Answer:** Choice B: $x + 10$ 21. **Problem:** Factor $x^2 + 7x + 6$. **Step 1:** Find factors of 6 that add to 7: $$ (x + 6)(x + 1) $$ 22. **Problem:** Factor $x^2 - 3x - 28$. **Step 1:** Find factors of -28 that add to -3: $$ (x - 7)(x + 4) $$ 23. **Problem:** Factor $10x^2 - x - 3$. **Step 1:** Multiply $10 \times (-3) = -30$. Find factors of -30 that add to -1: $$ 5 \text{ and } -6 $$ **Step 2:** Rewrite middle term: $$ 10x^2 + 5x - 6x - 3 $$ **Step 3:** Group: $$ (10x^2 + 5x) - (6x + 3) $$ **Step 4:** Factor each group: $$ 5x(2x + 1) - 3(2x + 1) $$ **Step 5:** Factor out common binomial: $$ (2x + 1)(5x - 3) $$ 24. **Problem:** Factor $15x^2 + 7x - 2$. **Step 1:** Multiply $15 \times (-2) = -30$. Find factors of -30 that add to 7: $$ 10 \text{ and } -3 $$ **Step 2:** Rewrite middle term: $$ 15x^2 + 10x - 3x - 2 $$ **Step 3:** Group: $$ (15x^2 + 10x) - (3x + 2) $$ **Step 4:** Factor each group: $$ 5x(3x + 2) - 1(3x + 2) $$ **Step 5:** Factor out common binomial: $$ (3x + 2)(5x - 1) $$ 25. **Problem:** Factor $x^2 - 25$. **Step 1:** Recognize difference of squares: $$ (x - 5)(x + 5) $$ 26. **Problem:** Factor $4x^2 - 81$. **Step 1:** Recognize difference of squares: $$ (2x - 9)(2x + 9) $$ 27. **Problem:** Factor $9x^2 - 12x + 4$. **Step 1:** Recognize perfect square trinomial: $$ (3x - 2)^2 $$ 28. **Problem:** Factor $16x^2 + 40x + 25$. **Step 1:** Recognize perfect square trinomial: $$ (4x + 5)^2 $$ 29. **Problem:** Solve $x^2 - 4x = 21$. **Step 1:** Rearrange: $$ x^2 - 4x - 21 = 0 $$ **Step 2:** Factor: $$ (x - 7)(x + 3) = 0 $$ **Step 3:** Solve: $$ x = 7 \quad \text{or} \quad x = -3 $$ 30. **Problem:** Solve $x^2 - 2x - 24 = 0$. **Step 1:** Factor: $$ (x - 6)(x + 4) = 0 $$ **Step 2:** Solve: $$ x = 6 \quad \text{or} \quad x = -4 $$ 31. **Problem:** Solve $6x^2 - 5x - 6 = 0$. **Step 1:** Multiply $6 \times (-6) = -36$. Find factors of -36 that add to -5: $$ 4 \text{ and } -9 $$ **Step 2:** Rewrite middle term: $$ 6x^2 + 4x - 9x - 6 = 0 $$ **Step 3:** Group: $$ (6x^2 + 4x) - (9x + 6) = 0 $$ **Step 4:** Factor each group: $$ 2x(3x + 2) - 3(3x + 2) = 0 $$ **Step 5:** Factor out common binomial: $$ (3x + 2)(2x - 3) = 0 $$ **Step 6:** Solve: $$ 3x + 2 = 0 \Rightarrow x = -\frac{2}{3} $$ $$ 2x - 3 = 0 \Rightarrow x = \frac{3}{2} $$ 32. **Problem:** Solve $2x^2 - 13x + 20 = 0$. **Step 1:** Multiply $2 \times 20 = 40$. Find factors of 40 that add to -13: $$ -8 \text{ and } -5 $$ **Step 2:** Rewrite middle term: $$ 2x^2 - 8x - 5x + 20 = 0 $$ **Step 3:** Group: $$ (2x^2 - 8x) - (5x - 20) = 0 $$ **Step 4:** Factor each group: $$ 2x(x - 4) - 5(x - 4) = 0 $$ **Step 5:** Factor out common binomial: $$ (x - 4)(2x - 5) = 0 $$ **Step 6:** Solve: $$ x - 4 = 0 \Rightarrow x = 4 $$ $$ 2x - 5 = 0 \Rightarrow x = \frac{5}{2} $$ 33. **Problem:** Factor $x^4 - 1$ completely and identify a factor. **Step 1:** Recognize difference of squares: $$ x^4 - 1 = (x^2)^2 - 1^2 = (x^2 - 1)(x^2 + 1) $$ **Step 2:** Factor $x^2 - 1$ further: $$ (x - 1)(x + 1)(x^2 + 1) $$ **Answer:** Choice G: $x - 1$ is a factor