\(\left(\frac{5y+1}{y^2-5y}+\frac{5y-1}{y^2+5y}\right).\frac{y^2-25}{y^2+1}\)
\(=\left(\frac{5y+1}{y\left(y-5\right)}+\frac{5y-1}{y\left(y+5\right)}\right).\frac{\left(y+5\right)\left(y-5\right)}{y^2+1}\)
\(=\left(\frac{\left(5y+1\right).\left(y+5\right)+\left(5y-1\right).\left(y-5\right)}{y\left(y-5\right)\left(y+5\right)}\right).\frac{\left(y+5\right)\left(y-5\right)}{y^2+1}\)
\(=\left(\frac{5y^2+25y+y+5+5y^2-25y-y+5}{y\left(y-5\right)\left(y+5\right)}\right).\frac{\left(y+5\right)\left(y-5\right)}{y^2+1}\)
\(=\frac{10y^2+10}{y\left(y-5\right)\left(y+5\right)}.\frac{\left(y+5\right)\left(y-5\right)}{y^2+1}\)
\(=\frac{10\left(y^2+1\right)}{y\left(y-5\right)\left(y+5\right)}.\frac{\left(y+5\right)\left(y-5\right)}{y^2+1}\)
\(=10\)