Answer
$\dfrac{(y^n+5)^2}{2(y^{n}+2)}$
Work Step by Step
Multiplying by the reciprocal of the divisor, the given expression, $
\dfrac{y^{2n}+7y^{n}+10}{10}\div\dfrac{y^{2n}+4y^n+4}{5y^n+25}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{y^{2n}+7y^{n}+10}{10}\cdot\dfrac{5y^n+25}{y^{2n}+4y^n+4}
.\end{array}
Factoring the expressions and then cancelling the common factor/s between the numerator and the denominator, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{(y^n+5)(y^n+2)}{10}\cdot\dfrac{5(y^n+5)}{(y^{n}+2)(y^{n}+2)}
\\=
\dfrac{(y^n+5)(\cancel{y^n+2})}{\cancel{5}(2)}\cdot\dfrac{\cancel5(y^n+5)}{(\cancel{y^n+2})(y^{n}+2)}
\\=
\dfrac{(y^n+5)^2}{2(y^{n}+2)}
.\end{array}
