perform the indicated operation of multiplication or division on the rational expressions and simplify…

Explanation:

Step1: Factor the quadratic expressions

Factor $y^{2}+12y + 36=(y + 6)^{2}$ using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$ where $a = y$ and $b = 6$. Factor $5y^{2}+34y + 24=5y^{2}+10y+24y + 24=5y(y + 2)+12(y + 2)=(5y + 12)(y+2)$. Factor $5y^{2}-11y - 12=5y^{2}-15y + 4y-12=5y(y - 3)+4(y - 3)=(5y + 4)(y - 3)$.

Step2: Rewrite the original expression

The original expression $\frac{y^{2}+12y + 36}{5y^{2}+34y + 24}\cdot\frac{5y^{2}-11y - 12}{y + 6}$ becomes $\frac{(y + 6)^{2}}{(5y + 12)(y + 2)}\cdot\frac{(5y + 4)(y - 3)}{y + 6}$.

Step3: Simplify the expression

Cancel out the common factor $(y + 6)$: $\frac{(y + 6)(y + 6)}{(5y + 12)(y + 2)}\cdot\frac{(5y + 4)(y - 3)}{y + 6}=\frac{(y + 6)(5y + 4)(y - 3)}{(5y + 12)(y + 2)}$.

Answer:

$\frac{(y + 6)(5y + 4)(y - 3)}{(5y + 12)(y + 2)}$