Simple fraction calculation

I'm trying to write this expression as one fraction:

$\begin{array}{rl}\frac{5}{y+1}\cdot \frac{1}{5}+\frac{y}{\frac{y+1}{3}}& =\frac{5y}{5y+5}+\frac{3y}{y+1}\\ & =\frac{5{y}^{2}+5+15{y}^{2}+15y}{5{y}^{2}+10y+5}\\ & =\frac{20{y}^{2}+15y+5}{5{y}^{2}+10y+5}\\ & =\frac{4{y}^{2}+3y+1}{2y}.\end{array}$

Am I doing this in the right way? Any hints about what to do next, please?

I'm trying to write this expression as one fraction:

$\begin{array}{rl}\frac{5}{y+1}\cdot \frac{1}{5}+\frac{y}{\frac{y+1}{3}}& =\frac{5y}{5y+5}+\frac{3y}{y+1}\\ & =\frac{5{y}^{2}+5+15{y}^{2}+15y}{5{y}^{2}+10y+5}\\ & =\frac{20{y}^{2}+15y+5}{5{y}^{2}+10y+5}\\ & =\frac{4{y}^{2}+3y+1}{2y}.\end{array}$

Am I doing this in the right way? Any hints about what to do next, please?