Question

fully simplify the expression below and write your answer as a single fraction.
\frac{x^{2}+3x - 10}{6(x + 5)^{2}}cdot\frac{x^{4}+3x^{3}-10x^{2}}{x(x - 2)^{2}}

Explanation:

Step1: Factor the polynomials

Factor $x^{2}+3x - 10=(x + 5)(x-2)$, $x^{4}+3x^{3}-10x^{2}=x^{2}(x^{2}+3x - 10)=x^{2}(x + 5)(x - 2)$.

Step2: Rewrite the original expression

The original expression $\frac{x^{2}+3x - 10}{6(x + 5)^{2}}\cdot\frac{x^{4}+3x^{3}-10x^{2}}{x(x - 2)^{2}}$ becomes $\frac{(x + 5)(x - 2)}{6(x + 5)^{2}}\cdot\frac{x^{2}(x + 5)(x - 2)}{x(x - 2)^{2}}$.

Step3: Cancel out the common factors

Cancel out the common factors: $(x + 5)$ terms, $(x - 2)$ terms and $x$ terms.
The $(x + 5)$ in the numerator and denominator: one $(x + 5)$ from $(x + 5)(x - 2)$ and one from $(x + 5)^{2}$, and one from $x^{2}(x + 5)(x - 2)$; one $(x - 2)$ from $(x + 5)(x - 2)$ and one from $x^{2}(x + 5)(x - 2)$ and one from $x(x - 2)^{2}$; one $x$ from $x^{2}(x + 5)(x - 2)$ and $x(x - 2)^{2}$.
We get $\frac{x}{6}$.

Answer:

$\frac{x}{6}$