Rewrite the expression with a common denominator.
\[\frac{2(y+1)(y+2)+1-{(y+2)}^{2}}{{(y+1)}^{2}{(y+2)}^{2}}\]
Expand.
\[\frac{2{y}^{2}+4y+2y+4+1-{y}^{2}-2y\times 2-{2}^{2}}{{(y+1)}^{2}{(y+2)}^{2}}\]
Simplify \({2}^{2}\) to \(4\).
\[\frac{2{y}^{2}+4y+2y+4+1-{y}^{2}-2y\times 2-4}{{(y+1)}^{2}{(y+2)}^{2}}\]
Simplify \(2y\times 2\) to \(4y\).
\[\frac{2{y}^{2}+4y+2y+4+1-{y}^{2}-4y-4}{{(y+1)}^{2}{(y+2)}^{2}}\]
Collect like terms.
\[\frac{(2{y}^{2}-{y}^{2})+(4y+2y-4y)+(4+1-4)}{{(y+1)}^{2}{(y+2)}^{2}}\]
Simplify \((2{y}^{2}-{y}^{2})+(4y+2y-4y)+(4+1-4)\) to \({y}^{2}+2y+1\).
\[\frac{{y}^{2}+2y+1}{{(y+1)}^{2}{(y+2)}^{2}}\]
Rewrite \({y}^{2}+2y+1\) in the form \({a}^{2}+2ab+{b}^{2}\), where \(a=y\) and \(b=1\).
\[\frac{{y}^{2}+2(y)(1)+{1}^{2}}{{(y+1)}^{2}{(y+2)}^{2}}\]
Use Square of Sum: \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\).
\[\frac{{(y+1)}^{2}}{{(y+1)}^{2}{(y+2)}^{2}}\]
Cancel \({(y+1)}^{2}\).
\[\frac{1}{{(y+2)}^{2}}\]
1/(y+2)^2
