Question

fully simplify the expression below and write your answer as a single fraction.
$\frac{6x^{4}-384x^{2}}{x + 3}cdot\frac{x^{2}-9}{6x^{2}+30x - 144}$

Explanation:

Step1: Factor the numerators and denominators

• Factor $6x^{4}-384x^{2}=6x^{2}(x^{2} - 64)=6x^{2}(x + 8)(x - 8)$
• Factor $x^{2}-9=(x + 3)(x - 3)$
• Factor $6x^{2}+30x - 144=6(x^{2}+5x - 24)=6(x + 8)(x - 3)$

Step2: Rewrite the original expression

The original expression $\frac{6x^{4}-384x^{2}}{x + 3}\cdot\frac{x^{2}-9}{6x^{2}+30x - 144}$ becomes $\frac{6x^{2}(x + 8)(x - 8)}{x + 3}\cdot\frac{(x + 3)(x - 3)}{6(x + 8)(x - 3)}$

Step3: Cancel out the common factors

Cancel out the common factors $6$, $(x + 8)$, $(x + 3)$ and $(x - 3)$ in the numerator and denominator.
We get $x^{2}(x - 8)$ which can be written as $\frac{x^{3}-8x^{2}}{1}$

Answer:

$x^{3}-8x^{2}$