Question

1. $\frac{3y + 2}{y^{2}+5y - 24}+\frac{7}{y^{2}+4y - 32}$

Explanation:

Step1: Factor the denominators

Factor $y^{2}+5y - 24=(y + 8)(y-3)$ and $y^{2}+4y - 32=(y + 8)(y - 4)$.
So the expression becomes $\frac{3y + 2}{(y + 8)(y-3)}+\frac{7}{(y + 8)(y - 4)}$.

Step2: Find the common - denominator

The common denominator is $(y + 8)(y-3)(y - 4)$.
Rewrite the fractions with the common denominator:
$\frac{(3y + 2)(y - 4)}{(y + 8)(y-3)(y - 4)}+\frac{7(y - 3)}{(y + 8)(y - 3)(y - 4)}$.

Step3: Expand the numerators

Expand $(3y + 2)(y - 4)=3y^{2}-12y+2y - 8=3y^{2}-10y - 8$ and $7(y - 3)=7y-21$.
The expression is $\frac{3y^{2}-10y - 8+7y - 21}{(y + 8)(y-3)(y - 4)}$.

Step4: Combine like - terms in the numerator

Combine like - terms: $3y^{2}-10y + 7y-8 - 21=3y^{2}-3y - 29$.
So the result is $\frac{3y^{2}-3y - 29}{(y + 8)(y-3)(y - 4)}$.

Answer:

$\frac{3y^{2}-3y - 29}{(y + 8)(y-3)(y - 4)}$