Answer
$ \frac{y^2+z^2}{2y-z^2}$.
Work Step by Step
The given expression is
$\Rightarrow \frac{y^3+y^2+yz^2+z^2}{y^3+y+y^2+1}\cdot \frac{y^3+y+y^2z+z}{2y^2+2yz-yz^2-z^3}$
Factor each numerator and denominator as shown below.
$\Rightarrow y^3+y^2+yz^2+z^2$
Group terms.
$\Rightarrow (y^3+y^2)+(yz^2+z^2)$
Factor each group.
$\Rightarrow y^2(y+1)+z^2(y+1)$
Factor out $(y+1)$.
$\Rightarrow (y+1)(y^2+z^2)$
$\Rightarrow y^3+y+y^2+1$
Group terms.
$\Rightarrow (y^3+y)+(y^2+1)$
Factor each group.
$\Rightarrow y(y^2+1)+1(y^2+1)$
Factor out $(y^2+1)$.
$\Rightarrow (y^2+1)(y+1)$
$\Rightarrow y^3+y+y^2z+z$
Group terms.
$\Rightarrow (y^3+y)+(y^2z+z)$
Factor each group.
$\Rightarrow y(y^2+1)+z(y^2+1)$
Factor out $(y^2+1)$.
$\Rightarrow (y^2+1)(y+z)$
$\Rightarrow 2y^2+2yz-yz^2-z^3$
Group terms.
$\Rightarrow (2y^2+2yz)+(-yz^2-z^3)$
Factor each group.
$\Rightarrow 2y(y+z)-z^2(y+z)$
Factor out $(y+z)$.
$\Rightarrow (y+z)(2y-z^2)$
Substitute all the factors into the given expression.
$\Rightarrow \frac{(y+1)(y^2+z^2)}{(y^2+1)(y+1)}\cdot \frac{(y^2+1)(y+z)}{(y+z)(2y-z^2)}$
Cancel common terms.
$\Rightarrow \frac{y^2+z^2}{2y-z^2}$.
