Question

fully simplify the expression below and write your answer as a single fraction.
\\(\frac{6x^{2}-384}{2x^{2}-36x + 160}\cdot\frac{x - 10}{x^{2}+12x + 32}\\)

Explanation:

Step1: Factor the numerators and denominators

First, factor out the greatest - common factor from $6x^{2}-384$:
$6x^{2}-384 = 6(x^{2}-64)=6(x + 8)(x - 8)$
Factor $2x^{2}-36x + 160$:
$2x^{2}-36x + 160=2(x^{2}-18x + 80)=2(x - 8)(x - 10)$
Factor $x^{2}+12x + 32$:
$x^{2}+12x + 32=(x + 4)(x + 8)$

Step2: Rewrite the original expression with factored forms

The original expression $\frac{6x^{2}-384}{2x^{2}-36x + 160}\cdot\frac{x - 10}{x^{2}+12x + 32}$ becomes $\frac{6(x + 8)(x - 8)}{2(x - 8)(x - 10)}\cdot\frac{x - 10}{(x + 4)(x + 8)}$

Step3: Cancel out the common factors

Cancel out the common factors $(x + 8)$, $(x - 8)$ and $(x - 10)$:
$\frac{6}{2}\cdot\frac{1}{x + 4}=\frac{3}{x + 4}$

Answer:

$\frac{3}{x + 4}$