Answer
$v=-\frac{4}{5} , -2$
Work Step by Step
We simplify as follows:
$(\frac{v}{v+1})^{2}+\frac{2v}{v+1}=8$
$\frac{v^{2}}{(v+1)^{2}}+\frac{2v(v+1)}{(v+1)^{2}}=\frac{8(v+1)^{2}}{(v+1)^{2}}$
$v^{2}+2v(v+1)=8(v+1)^{2}$
$3v^{2}+2v=8v^{2}+16v+8$
$5v^{2}+14v+8=0$
$(5v+4)(v+2)=0$
$v=-\frac{4}{5}, -2$