Question

perform the following operation and express in simplest form.\\(dfrac{9x + 72}{x^2 + 10x + 16} cdot dfrac{x^2 - 7x - 18}{x - 9}\\)

Explanation:

Step1: Factor all polynomials

• Numerator 1: $9x + 72 = 9(x + 8)$
• Denominator 1: $x^2 + 10x + 16 = (x + 2)(x + 8)$
• Numerator 2: $x^2 - 7x - 18 = (x - 9)(x + 2)$
• Denominator 2: $x - 9$

Substitute into the expression:
$$\frac{9(x + 8)}{(x + 2)(x + 8)} \cdot \frac{(x - 9)(x + 2)}{x - 9}$$

Step2: Cancel common factors

Cancel $(x+8)$, $(x+2)$, and $(x-9)$ from numerator and denominator:
$$\frac{9\cancel{(x + 8)}}{\cancel{(x + 2)}\cancel{(x + 8)}} \cdot \frac{\cancel{(x - 9)}\cancel{(x + 2)}}{\cancel{x - 9}}$$

Answer:

$9$ (with restrictions $x
eq -8, -2, 9$)