1. Solve for $x$ in the equation $x - 5 = -8$. $$x - 5 = -8$$ Add 5 to both sides: $$x - 5 + 5 = -8 + 5$$ $$x = -3$$ 2. Evaluate $(x^2 - 5)(x^{-3} - x)$ at $x = -3$. Calculate each part: $$x^2 - 5 = (-3)^2 - 5 = 9 - 5 = 4$$ $$x^{-3} - x = (-3)^{-3} - (-3) = \frac{1}{(-3)^3} + 3 = \frac{1}{-27} + 3 = -\frac{1}{27} + 3 = \frac{-1 + 81}{27} = \frac{80}{27}$$ Multiply: $$(x^2 - 5)(x^{-3} - x) = 4 \times \frac{80}{27} = \frac{320}{27}$$ 3. Factor and simplify each quadratic: 4. $x^2 + 6x - 16$ Try factors of -16 that sum to 6: 8 and -2. $$x^2 + 6x - 16 = (x + 8)(x - 2)$$ 5. $x^2 - 6x + 9$ Try factors of 9 that sum to -6: -3 and -3. $$x^2 - 6x + 9 = (x - 3)^2$$ 6. $x^2 - 6x - 27$ Try factors of -27 that sum to -6: -9 and 3. $$x^2 - 6x - 27 = (x - 9)(x + 3)$$ 7. $p^2 - p - 20$ Factors of -20 that sum to -1: -5 and 4. $$p^2 - p - 20 = (p - 5)(p + 4)$$ 8. $x^2 - 2x - 15$ Factors of -15 that sum to -2: -5 and 3. $$x^2 - 2x - 15 = (x - 5)(x + 3)$$ 9. $p^2 - 4p - 21$ Factors of -21 that sum to -4: -7 and 3. $$p^2 - 4p - 21 = (p - 7)(p + 3)$$ 10. $p^2 + p - 20$ Factors of -20 that sum to 1: 5 and -4. $$p^2 + p - 20 = (p + 5)(p - 4)$$ 11. $k^2 - 3k - 40$ Factors of -40 that sum to -3: -8 and 5. $$k^2 - 3k - 40 = (k - 8)(k + 5)$$ 12. $m^2 + 9m + 20$ Factors of 20 that sum to 9: 5 and 4. $$m^2 + 9m + 20 = (m + 5)(m + 4)$$ 13. $x^2 + 33 + 14x$ Rewrite as $x^2 + 14x + 33$. Factors of 33 that sum to 14: 11 and 3. $$x^2 + 14x + 33 = (x + 11)(x + 3)$$ 14. $-13p + p^2 + 36$ Rewrite as $p^2 - 13p + 36$. Factors of 36 that sum to -13: -9 and -4. $$p^2 - 13p + 36 = (p - 9)(p - 4)$$ 15. $-30 + m^2 - m$ Rewrite as $m^2 - m - 30$. Factors of -30 that sum to -1: -6 and 5. $$m^2 - m - 30 = (m - 6)(m + 5)$$ 16. $11n + n^2 + 18$ Rewrite as $n^2 + 11n + 18$. Factors of 18 that sum to 11: 9 and 2. $$n^2 + 11n + 18 = (n + 9)(n + 2)$$ 17. $x^2 + 27 + 12x$ Rewrite as $x^2 + 12x + 27$. Factors of 27 that sum to 12: 9 and 3. $$x^2 + 12x + 27 = (x + 9)(x + 3)$$ 18. $x^2 + 90 - 19x$ Rewrite as $x^2 - 19x + 90$. Factors of 90 that sum to -19: -10 and -9. $$x^2 - 19x + 90 = (x - 10)(x - 9)$$ 19. $x^2 + x - 132$ Factors of -132 that sum to 1: 12 and -11. $$x^2 + x - 132 = (x + 12)(x - 11)$$ 20. $a^2 + 90 - 47a$ Rewrite as $a^2 - 47a + 90$. Factors of 90 that sum to -47: -45 and -2. $$a^2 - 47a + 90 = (a - 45)(a - 2)$$ 21. $10m + m^2 + 16$ Rewrite as $m^2 + 10m + 16$. Factors of 16 that sum to 10: 8 and 2. $$m^2 + 10m + 16 = (m + 8)(m + 2)$$ 22. Solve system: $$\begin{cases} 3x + y = 9 \\ x - 4y = -10 \end{cases}$$ Multiply second by 3: $$3x - 12y = -30$$ Subtract first: $$(3x - 12y) - (3x + y) = -30 - 9$$ $$-13y = -39$$ $$y = 3$$ Substitute $y=3$ into $3x + y = 9$: $$3x + 3 = 9$$ $$3x = 6$$ $$x = 2$$ 23. Solve system: $$\begin{cases} 2x + 5y = 7 \\ x + 3y = 4 \end{cases}$$ Multiply second by 2: $$2x + 6y = 8$$ Subtract first: $$(2x + 6y) - (2x + 5y) = 8 - 7$$ $$y = 1$$ Substitute $y=1$ into $x + 3y = 4$: $$x + 3 = 4$$ $$x = 1$$ 24. Solve system: $$\begin{cases} 3x + 4y = -7 \\ 3x - 3y = 21 \end{cases}$$ Subtract second from first: $$(3x + 4y) - (3x - 3y) = -7 - 21$$ $$7y = -28$$ $$y = -4$$ Substitute $y=-4$ into $3x + 4y = -7$: $$3x + 4(-4) = -7$$ $$3x - 16 = -7$$ $$3x = 9$$ $$x = 3$$ 25. Solve system: $$\begin{cases} 2x - 2y = -2 \\ 4x - 5y = -9 \end{cases}$$ Multiply first by 2: $$4x - 4y = -4$$ Subtract from second: $$(4x - 5y) - (4x - 4y) = -9 - (-4)$$ $$-y = -5$$ $$y = 5$$ Substitute $y=5$ into $2x - 2y = -2$: $$2x - 2(5) = -2$$ $$2x - 10 = -2$$ $$2x = 8$$ $$x = 4$$ 26. Simplify $7\sqrt{20} - 5\sqrt{32}$. $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$$ Substitute: $$7 \times 2\sqrt{5} - 5 \times 4\sqrt{2} = 14\sqrt{5} - 20\sqrt{2}$$ 27. Simplify $2\sqrt{18} - 5\sqrt{8} + 4\sqrt{50}$. $$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ Substitute: $$2 \times 3\sqrt{2} - 5 \times 2\sqrt{2} + 4 \times 5\sqrt{2} = 6\sqrt{2} - 10\sqrt{2} + 20\sqrt{2}$$ Combine: $$(6 - 10 + 20)\sqrt{2} = 16\sqrt{2}$$ 28. Simplify $\frac{1 + \frac{1}{y}}{\frac{1}{y}}$. Rewrite numerator: $$1 + \frac{1}{y} = \frac{y}{y} + \frac{1}{y} = \frac{y + 1}{y}$$ Divide: $$\frac{\frac{y + 1}{y}}{\frac{1}{y}} = \frac{y + 1}{y} \times \frac{y}{1} = y + 1$$ 29. Simplify $\frac{\frac{a}{b} - 4}{\frac{x}{b} - b}$. Rewrite denominator: $$\frac{x}{b} - b = \frac{x}{b} - \frac{b^2}{b} = \frac{x - b^2}{b}$$ Rewrite numerator: $$\frac{a}{b} - 4 = \frac{a}{b} - \frac{4b}{b} = \frac{a - 4b}{b}$$ Divide: $$\frac{\frac{a - 4b}{b}}{\frac{x - b^2}{b}} = \frac{a - 4b}{b} \times \frac{b}{x - b^2} = \frac{a - 4b}{x - b^2}$$ 30. Simplify $\frac{\frac{a}{x} - a}{x + \frac{y}{x}}$. Rewrite numerator: $$\frac{a}{x} - a = \frac{a}{x} - \frac{ax}{x} = \frac{a - ax}{x} = \frac{a(1 - x)}{x}$$ Rewrite denominator: $$x + \frac{y}{x} = \frac{x^2}{x} + \frac{y}{x} = \frac{x^2 + y}{x}$$ Divide: $$\frac{\frac{a(1 - x)}{x}}{\frac{x^2 + y}{x}} = \frac{a(1 - x)}{x} \times \frac{x}{x^2 + y} = \frac{a(1 - x)}{x^2 + y}$$