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

First fraction:

Numerator: $x^2 + 14x + 40 = (x+4)(x+10)$
Denominator: $9x^4 + 135x^3 + 450x^2 = 9x^2(x^2 + 15x + 50) = 9x^2(x+5)(x+10)$

Second fraction:

Numerator: $6x^2 + 84x + 270 = 6(x^2 + 14x + 45) = 6(x+5)(x+9)$
Denominator: $x^2 + 13x + 36 = (x+4)(x+9)$

Substitute back into the product:
$$\frac{(x+4)(x+10)}{9x^2(x+5)(x+10)} \cdot \frac{6(x+5)(x+9)}{(x+4)(x+9)}$$

Step2: Cancel common factors

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

Step3: Multiply remaining terms

Multiply the numerators and denominators, then simplify the constant fraction:
$$\frac{6}{9x^2} = \frac{2}{3x^2}$$

Answer:

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