Question

fully simplify the expression below and write your answer as a single fraction.
\frac{x^{2}+14x + 40}{9x^{4}+135x^{3}+450x^{2}} cdot \frac{6x^{2}+84x + 270}{x^{2}+13x + 36}

Explanation:

Step1: Factor the quadratic expressions

$x^{2}+14x + 40=(x + 4)(x+10)$; $6x^{2}+84x + 270=6(x^{2}+14x + 45)=6(x + 5)(x + 9)$; $x^{2}+13x + 36=(x + 4)(x+9)$; $9x^{4}+135x^{3}+450x^{2}=9x^{2}(x^{2}+15x + 50)=9x^{2}(x + 5)(x + 10)$

Step2: Substitute the factored - forms into the original expression

The original expression $\frac{x^{2}+14x + 40}{9x^{4}+135x^{3}+450x^{2}}\cdot\frac{6x^{2}+84x + 270}{x^{2}+13x + 36}$ becomes $\frac{(x + 4)(x + 10)}{9x^{2}(x + 5)(x + 10)}\cdot\frac{6(x + 5)(x + 9)}{(x + 4)(x + 9)}$

Step3: Cancel out the common factors

Cancel out the common factors $(x + 4)$, $(x + 5)$, $(x + 9)$ and $(x + 10)$ in the numerator and the denominator.
We have $\frac{1}{9x^{2}}\cdot\frac{6}{1}=\frac{6}{9x^{2}}=\frac{2}{3x^{2}}$

Answer:

$\frac{2}{3x^{2}}$