Question: Simplify each rational expression. 1. $\frac{5y^2 + 11y + 2}{5y^2 + y}$ 2. $\frac{20 - 5x}{3x^2 - 48}$ 3. $\frac{8k - 10k - 3}{6k^2 - 13k + 6}$ PRACTICE Simplify each rational expression. 1. $\frac{-4n^2 + 4n - n^2}{2u^2 + uv - v^2}$ 2. $\frac{x - 7}{(6x - 4)(x - 7)}$ 3. $\frac{x(x^3 + 7x^2) + 2x^2(x + 9)}{x^4 - 2(x^3 + 7x^2) - x^2}$ 4. $\frac{-108p^2 - 153p - 54}{96p^2 + 104p + 24}$ Factor the practice with grade 8 techniques and in a simpler form that is easy to understand.

1. **Problem:** Simplify $\frac{-4n^2 + 4n - n^2}{2u^2 + uv - v^2}$. **Step 1:** Combine like terms in numerator: $$-4n^2 - n^2 + 4n = -5n^2 + 4n$$ **Step 2:** Factor numerator: $$-5n^2 + 4n = n(-5n + 4)$$ **Step 3:** Factor denominator $2u^2 + uv - v^2$ by grouping or trial: Find two numbers that multiply to $2 \times (-1) = -2$ and add to $1$ (coefficient of $uv$): these are $2$ and $-1$. Rewrite: $$2u^2 + 2uv - uv - v^2 = 2u(u + v) - v(u + v) = (2u - v)(u + v)$$ **Step 4:** Final simplified form: $$\frac{n(-5n + 4)}{(2u - v)(u + v)}$$ 2. **Problem:** Simplify $\frac{x - 7}{(6x - 4)(x - 7)}$. **Step 1:** Cancel common factor $(x - 7)$: $$\frac{\cancel{x - 7}}{(6x - 4)\cancel{(x - 7)}} = \frac{1}{6x - 4}$$ **Step 2:** Factor denominator: $$6x - 4 = 2(3x - 2)$$ **Step 3:** Final simplified form: $$\frac{1}{2(3x - 2)}$$ 3. **Problem:** Simplify $\frac{x(x^3 + 7x^2) + 2x^2(x + 9)}{x^4 - 2(x^3 + 7x^2) - x^2}$. **Step 1:** Expand numerator: $$x(x^3 + 7x^2) = x^4 + 7x^3$$ $$2x^2(x + 9) = 2x^3 + 18x^2$$ Sum numerator: $$x^4 + 7x^3 + 2x^3 + 18x^2 = x^4 + 9x^3 + 18x^2$$ **Step 2:** Expand denominator: $$x^4 - 2(x^3 + 7x^2) - x^2 = x^4 - 2x^3 - 14x^2 - x^2 = x^4 - 2x^3 - 15x^2$$ **Step 3:** Factor numerator: Factor out $x^2$: $$x^2(x^2 + 9x + 18)$$ Factor quadratic: $$x^2 + 9x + 18 = (x + 6)(x + 3)$$ So numerator: $$x^2(x + 6)(x + 3)$$ **Step 4:** Factor denominator: Factor out $x^2$: $$x^2(x^2 - 2x - 15)$$ Factor quadratic: $$x^2 - 2x - 15 = (x - 5)(x + 3)$$ So denominator: $$x^2(x - 5)(x + 3)$$ **Step 5:** Cancel common factors $x^2$ and $(x + 3)$: $$\frac{\cancel{x^2}(x + 6)\cancel{(x + 3)}}{\cancel{x^2}(x - 5)\cancel{(x + 3)}} = \frac{x + 6}{x - 5}$$ 4. **Problem:** Simplify $\frac{-108p^2 - 153p - 54}{96p^2 + 104p + 24}$. **Step 1:** Factor numerator: Factor out $-3$: $$-3(36p^2 + 51p + 18)$$ Factor quadratic inside: Find factors of $36 \times 18 = 648$ that add to $51$: $27$ and $24$. Rewrite: $$36p^2 + 27p + 24p + 18 = 9p(4p + 3) + 6(4p + 3) = (9p + 6)(4p + 3)$$ So numerator: $$-3(9p + 6)(4p + 3)$$ **Step 2:** Factor denominator: Factor out $4$: $$4(24p^2 + 26p + 6)$$ Factor quadratic inside: Find factors of $24 \times 6 = 144$ that add to $26$: $18$ and $8$. Rewrite: $$24p^2 + 18p + 8p + 6 = 6p(4p + 3) + 2(4p + 3) = (6p + 2)(4p + 3)$$ So denominator: $$4(6p + 2)(4p + 3)$$ **Step 3:** Simplify numerator factor $9p + 6$ and denominator factor $6p + 2$: $$9p + 6 = 3(3p + 2)$$ $$6p + 2 = 2(3p + 1)$$ (Note: no common factor with numerator's $3p + 2$) So numerator: $$-3 \times 3(3p + 2)(4p + 3) = -9(3p + 2)(4p + 3)$$ Denominator: $$4 \times 2(3p + 1)(4p + 3) = 8(3p + 1)(4p + 3)$$ **Step 4:** Cancel common factor $(4p + 3)$: $$\frac{-9(3p + 2)\cancel{(4p + 3)}}{8(3p + 1)\cancel{(4p + 3)}} = \frac{-9(3p + 2)}{8(3p + 1)}$$ **Final answers:** 1. $\frac{n(-5n + 4)}{(2u - v)(u + v)}$ 2. $\frac{1}{2(3x - 2)}$ 3. $\frac{x + 6}{x - 5}$ 4. $\frac{-9(3p + 2)}{8(3p + 1)}$