Question

fully simplify the expression below and write your answer as a single fraction.
$\frac{5x^{3}+20x^{2}-160x}{x - 6}cdot\frac{x^{2}-36}{15x^{4}+210x^{3}+720x^{2}}$

Explanation:

Step1: Factor the expressions

Factor the numerator of the first - fraction:
\[

$$\begin{align*} 5x^{3}+20x^{2}-160x&=5x(x^{2} + 4x-32)\\ &=5x(x + 8)(x - 4) \end{align*}$$

\]
Factor the denominator of the second - fraction:
\[

$$\begin{align*} 15x^{4}+210x^{3}+720x^{2}&=15x^{2}(x^{2}+14x + 48)\\ &=15x^{2}(x + 6)(x+8) \end{align*}$$

\]
Factor the numerator of the second - fraction: \(x^{2}-36=(x + 6)(x - 6)\)

Step2: Rewrite the original expression

The original expression \(\frac{5x^{3}+20x^{2}-160x}{x - 6}\cdot\frac{x^{2}-36}{15x^{4}+210x^{3}+720x^{2}}\) becomes \(\frac{5x(x + 8)(x - 4)}{x - 6}\cdot\frac{(x + 6)(x - 6)}{15x^{2}(x + 6)(x + 8)}\)

Step3: Cancel out the common factors

Cancel out the common factors \((x - 6)\), \((x + 6)\), \((x + 8)\) and simplify the coefficients and \(x\) terms.
\[

$$\begin{align*} &\frac{5x(x + 8)(x - 4)}{x - 6}\cdot\frac{(x + 6)(x - 6)}{15x^{2}(x + 6)(x + 8)}\\ =&\frac{5x(x - 4)}{15x^{2}}\\ =&\frac{x-4}{3x} \end{align*}$$

\]

Answer:

\(\frac{x - 4}{3x}\)