Answer
$\displaystyle \frac{10-y}{y-7}$
Work Step by Step
Step by step multiplication of rational expressions:
1. Factor completely what you can
2. Reduce (divide) numerators and denominators by common factors.
3. Multiply the remaining factors in the numerators and
multiply the remaining factors in the denominators. $(\displaystyle \frac{P}{Q}\cdot\frac{R}{S}=\frac{PR}{QS})$
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Factor what we can:
... difference of squares, factor out $(-1)$ first
$25-y^{2}=-(y^{2}-5^{5})=-(y+5)(y-5)$
... factor $y^{2}+by+c$ by searching for two factors of $c$ whose sum is $b$.
$y^{2}-8y-20=(y-10)(y+2)$
$y^{2}-2y-35=(y-7)(y+5)$
$y^{2}-3y-10=(y-5)(y+2)$
The problem becomes
$...=\displaystyle \frac{-(y+5)(y-5)\cdot(y-10)(y+2)}{(y-7)(y+5)\cdot(y-5)(y+2)}\qquad$ ... divide out the common factors
$=\displaystyle \frac{-\fbox{$(y+5)$}\fbox{$(y-5)$}\cdot(y-10)\fbox{$(y+2)$}}{(y-7)\fbox{$(y+5)$}\cdot\fbox{$(y-5)$}\fbox{$(y+2)$}}\qquad$
$-\displaystyle \frac{-(y-10)}{y-7}$
= $\displaystyle \frac{10-y}{y-7}$