Question

divide.
\\(\frac{7x^2 - 30x + 8}{x + 9} \div \frac{2x - 8}{5x - 8}\\)
simplify your answer as much as possible.

Explanation:

Step1: Rewrite division as multiplication

$\frac{7x^2 - 30x + 8}{x+9} \times \frac{5x-8}{2x-8}$

Step2: Factor all polynomials

Factor $7x^2-30x+8$: find two numbers that multiply to $7\times8=56$ and add to $-30$, which are $-28$ and $-2$.
$7x^2-28x-2x+8 = 7x(x-4)-2(x-4)=(7x-2)(x-4)$
Factor $2x-8$: $2(x-4)$
Substitute back:
$\frac{(7x-2)(x-4)}{x+9} \times \frac{5x-8}{2(x-4)}$

Step3: Cancel common factors

Cancel $(x-4)$ from numerator and denominator:
$\frac{(7x-2)}{x+9} \times \frac{5x-8}{2}$

Step4: Multiply numerators and denominators

$\frac{(7x-2)(5x-8)}{2(x+9)}$

Step5: Expand numerator (optional, simplified form)

$(7x-2)(5x-8)=35x^2-56x-10x+16=35x^2-66x+16$
So: $\frac{35x^2-66x+16}{2x+18}$

Answer:

$\frac{(7x-2)(5x-8)}{2(x+9)}$ or $\frac{35x^2-66x+16}{2x+18}$ (both are fully simplified)