1. **Problem:** Simplify the expression $(8a - 5a^3) + (4a^3 + a)$. 2. **Formula and rules:** Combine like terms by adding or subtracting coefficients of terms with the same variable and exponent. 3. **Work:** $$(8a - 5a^3) + (4a^3 + a) = 8a - 5a^3 + 4a^3 + a$$ Group like terms: $$= (8a + a) + (-5a^3 + 4a^3)$$ Simplify coefficients: $$= 9a - a^3$$ 4. **Answer:** $9a - a^3$ --- 1. **Problem:** Simplify the expression $(2a + 8 + 5a^2) - (a^2 - 4a) - (7a + 2)$. 2. **Formula and rules:** Distribute the minus signs and combine like terms. 3. **Work:** $$(2a + 8 + 5a^2) - (a^2 - 4a) - (7a + 2) = 2a + 8 + 5a^2 - a^2 + 4a - 7a - 2$$ Group like terms: $$= (5a^2 - a^2) + (2a + 4a - 7a) + (8 - 2)$$ Simplify coefficients: $$= 4a^2 - a + 6$$ 4. **Answer:** $4a^2 - a + 6$ --- 1. **Problem:** Simplify the expression $(3v + 3 + 5v^4) - (6 + 6v) - (4v^4 + 1 + 5v)$. 2. **Formula and rules:** Distribute minus signs and combine like terms. 3. **Work:** $$(3v + 3 + 5v^4) - (6 + 6v) - (4v^4 + 1 + 5v) = 3v + 3 + 5v^4 - 6 - 6v - 4v^4 - 1 - 5v$$ Group like terms: $$= (5v^4 - 4v^4) + (3v - 6v - 5v) + (3 - 6 - 1)$$ Simplify coefficients: $$= v^4 - 8v - 4$$ 4. **Answer:** $v^4 - 8v - 4$ --- 1. **Problem:** Simplify the expression $(7x^3 + 7 - x^2) - (3 + x + x^2 - 4x^3) - (3x^2 + 2 + 4x^3 - 5x)$. 2. **Formula and rules:** Distribute minus signs and combine like terms. 3. **Work:** $$(7x^3 + 7 - x^2) - (3 + x + x^2 - 4x^3) - (3x^2 + 2 + 4x^3 - 5x)$$ $$= 7x^3 + 7 - x^2 - 3 - x - x^2 + 4x^3 - 3x^2 - 2 - 4x^3 + 5x$$ Group like terms: $$= (7x^3 + 4x^3 - 4x^3) + (-x^2 - x^2 - 3x^2) + (-x + 5x) + (7 - 3 - 2)$$ Simplify coefficients: $$= 7x^3 - 5x^2 + 4x + 2$$ 4. **Answer:** $7x^3 - 5x^2 + 4x + 2$ --- 1. **Problem:** Simplify the expression $(5 - 2x^3 - 6x) - (2x^4 - 2x^2 + 5x^3 - 5) - (6x^3 + x^2 + 1 + 4x)$. 2. **Formula and rules:** Distribute minus signs and combine like terms. 3. **Work:** $$(5 - 2x^3 - 6x) - (2x^4 - 2x^2 + 5x^3 - 5) - (6x^3 + x^2 + 1 + 4x)$$ $$= 5 - 2x^3 - 6x - 2x^4 + 2x^2 - 5x^3 + 5 - 6x^3 - x^2 - 1 - 4x$$ Group like terms: $$= -2x^4 + (2x^2 - x^2) + (-2x^3 - 5x^3 - 6x^3) + (-6x - 4x) + (5 + 5 - 1)$$ Simplify coefficients: $$= -2x^4 + x^2 - 13x^3 - 10x + 9$$ 4. **Answer:** $-2x^4 + x^2 - 13x^3 - 10x + 9$ --- 1. **Problem:** Find the product $(8x + 2)(x - 2)$. 2. **Formula and rules:** Use distributive property (FOIL) to multiply binomials. 3. **Work:** $$ (8x + 2)(x - 2) = 8x \cdot x + 8x \cdot (-2) + 2 \cdot x + 2 \cdot (-2) $$ $$= 8x^2 - 16x + 2x - 4$$ Combine like terms: $$= 8x^2 - 14x - 4$$ 4. **Answer:** $8x^2 - 14x - 4$ --- 1. **Problem:** Find the product $(n + 7)(2n^2 - 4n + 4)$. 2. **Formula and rules:** Use distributive property to multiply each term. 3. **Work:** $$ (n + 7)(2n^2 - 4n + 4) = n(2n^2 - 4n + 4) + 7(2n^2 - 4n + 4) $$ $$= 2n^3 - 4n^2 + 4n + 14n^2 - 28n + 28$$ Combine like terms: $$= 2n^3 + 10n^2 - 24n + 28$$ 4. **Answer:** $2n^3 + 10n^2 - 24n + 28$ --- 1. **Problem:** Find the product $(3n + 3)(7n^2 - 2n + 2)$. 2. **Formula and rules:** Use distributive property. 3. **Work:** $$ (3n + 3)(7n^2 - 2n + 2) = 3n(7n^2 - 2n + 2) + 3(7n^2 - 2n + 2) $$ $$= 21n^3 - 6n^2 + 6n + 21n^2 - 6n + 6$$ Combine like terms: $$= 21n^3 + 15n^2 + 0n + 6 = 21n^3 + 15n^2 + 6$$ 4. **Answer:** $21n^3 + 15n^2 + 6$ --- 1. **Problem:** Find the product $(m - 3)(5m^2 + 4m + 1)$. 2. **Formula and rules:** Use distributive property. 3. **Work:** $$ (m - 3)(5m^2 + 4m + 1) = m(5m^2 + 4m + 1) - 3(5m^2 + 4m + 1) $$ $$= 5m^3 + 4m^2 + m - 15m^2 - 12m - 3$$ Combine like terms: $$= 5m^3 - 11m^2 - 11m - 3$$ 4. **Answer:** $5m^3 - 11m^2 - 11m - 3$ --- 1. **Problem:** Find the product $(6k + 4)(3k^2 + 6k - 5)$. 2. **Formula and rules:** Use distributive property. 3. **Work:** $$ (6k + 4)(3k^2 + 6k - 5) = 6k(3k^2 + 6k - 5) + 4(3k^2 + 6k - 5) $$ $$= 18k^3 + 36k^2 - 30k + 12k^2 + 24k - 20$$ Combine like terms: $$= 18k^3 + 48k^2 - 6k - 20$$ 4. **Answer:** $18k^3 + 48k^2 - 6k - 20$