Question

what is the quotient?\\(\frac{2y^{2}-6y - 20}{4y + 12}div\frac{y^{2}+5y + 6}{3y^{2}+18y + 27}\\)\\(\bigcirc\frac{2}{3(y - 5)}\\)\\(\bigcirc\frac{3(y - 5)}{2}\\)\\(\bigcirc\frac{(y - 5)(y + 2)^{2}}{8(y + 3)^{2}}\\)\\(\bigcirc\frac{3(y + 5)(y - 2)}{2(y + 2)}\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{2y^2-6y-20}{4y+12} \times \frac{3y^2+18y+27}{y^2+5y+6}$

Step2: Factor all polynomials

• Numerator 1: $2(y^2-3y-10)=2(y-5)(y+2)$
• Denominator 1: $4(y+3)$
• Numerator 2: $3(y^2+6y+9)=3(y+3)^2$
• Denominator 2: $y^2+5y+6=(y+2)(y+3)$

Substitute back:
$\frac{2(y-5)(y+2)}{4(y+3)} \times \frac{3(y+3)^2}{(y+2)(y+3)}$

Step3: Cancel common factors

Cancel $(y+2)$, $(y+3)$ from numerator/denominator; simplify $\frac{2}{4}=\frac{1}{2}$:
$\frac{(y-5)}{2} \times \frac{3(y+3)}{(y+3)}$
Cancel remaining $(y+3)$:
$\frac{3(y-5)}{2}$

Answer:

$\boldsymbol{\frac{3(y-5)}{2}}$ (matches the second option)