Question

perform the operation and simplify.\\(\frac{x^2 + 3x + 2}{x^2 + 7x + 12} \div \frac{x^2 - 4x - 5}{x^2 - 3x - 28}\\)\\(\frac{(x - ?)(x + \square)}{(x - \square)(x + \square)} +\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{x^2 + 3x + 2}{x^2 + 7x + 12} \times \frac{x^2 - 3x - 28}{x^2 - 4x - 5}$

Step2: Factor all quadratics

$\frac{(x+1)(x+2)}{(x+3)(x+4)} \times \frac{(x-7)(x+4)}{(x-5)(x+1)}$

Step3: Cancel common factors

Cancel $(x+1)$ and $(x+4)$ from numerator/denominator:
$\frac{(x+2)}{(x+3)} \times \frac{(x-7)}{(x-5)}$

Step4: Multiply remaining terms

$\frac{(x-7)(x+2)}{(x-5)(x+3)}$

Answer:

$\frac{(x-7)(x+2)}{(x-5)(x+3)}$
(Filling the blanks: green box = 7, top beige box = 2, bottom left beige box = 5, bottom right beige box = 3)