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 all numerators/denominators

Numerator 1: $x^2+14x+40=(x+4)(x+10)$
Denominator 1: $9x^4+135x^3+450x^2=9x^2(x^2+15x+50)=9x^2(x+5)(x+10)$
Numerator 2: $6x^2+84x+270=6(x^2+14x+45)=6(x+5)(x+9)$
Denominator 2: $x^2+13x+36=(x+4)(x+9)$

Step2: Rewrite product with factors

$$\frac{(x+4)(x+10)}{9x^2(x+5)(x+10)} \cdot \frac{6(x+5)(x+9)}{(x+4)(x+9)}$$

Step3: Cancel common factors

Cancel $(x+4), (x+10), (x+5), (x+9)$ from numerator/denominator:
$$\frac{1}{9x^2} \cdot \frac{6}{1}$$

Step4: Multiply remaining terms

$$\frac{6}{9x^2} = \frac{2}{3x^2}$$

Answer:

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