Question

simplify each. make sure to reduce the fractions\\
\\(\frac{x^2 - x - 6}{x^2 + 5x + 6} \cdot \frac{4x + 12}{x^2 - 8x + 15}\\)\\(=\\)\\
\\(\frac{2x - 4}{x^2 - 5x + 4} \div \frac{x^2 - 7x + 10}{x^2 - 9x + 20}\\)\\(=\\)\\
question help: \\(\triangledown\\) video\\
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Explanation:

Step1: Factor all polynomials (First problem)

$\frac{(x-3)(x+2)}{(x+2)(x+3)} \cdot \frac{4(x+3)}{(x-3)(x-5)}$

Step2: Cancel common factors

$\frac{\cancel{(x-3)}\cancel{(x+2)}}{\cancel{(x+2)}\cancel{(x+3)}} \cdot \frac{4\cancel{(x+3)}}{\cancel{(x-3)}(x-5)}$

Step3: Simplify remaining terms

$\frac{4}{x-5}$

Step1: Rewrite division as multiplication (Second problem)

$\frac{2x-4}{x^2-5x+4} \cdot \frac{x^2-9x+20}{x^2-7x+10}$

Step2: Factor all polynomials

$\frac{2(x-2)}{(x-1)(x-4)} \cdot \frac{(x-4)(x-5)}{(x-2)(x-5)}$

Step3: Cancel common factors

$\frac{2\cancel{(x-2)}}{(x-1)\cancel{(x-4)}} \cdot \frac{\cancel{(x-4)}\cancel{(x-5)}}{\cancel{(x-2)}\cancel{(x-5)}}$

Step4: Simplify remaining terms

$\frac{2}{x-1}$

Answer:

First simplified expression: $\frac{4}{x-5}$
Second simplified expression: $\frac{2}{x-1}$