$$\frac { ( 2 y + 2 ) ^ { 2 } } { 4 } - \frac { 3 ( y + 2 ) ^ { 2 } } { 6 } = \frac { y ( y - 2 ) } { 2 }$$
$y=1$
$$3\left(2y+2\right)^{2}-2\times 3\left(y+2\right)^{2}=6y\left(y-2\right)$$
$$3\left(4y^{2}+8y+4\right)-2\times 3\left(y+2\right)^{2}=6y\left(y-2\right)$$
$$12y^{2}+24y+12-2\times 3\left(y+2\right)^{2}=6y\left(y-2\right)$$
$$12y^{2}+24y+12-6\left(y+2\right)^{2}=6y\left(y-2\right)$$
$$12y^{2}+24y+12-6\left(y^{2}+4y+4\right)=6y\left(y-2\right)$$
$$12y^{2}+24y+12-6y^{2}-24y-24=6y\left(y-2\right)$$
$$6y^{2}+24y+12-24y-24=6y\left(y-2\right)$$
$$6y^{2}+12-24=6y\left(y-2\right)$$
$$6y^{2}-12=6y\left(y-2\right)$$
$$6y^{2}-12=6y^{2}-12y$$
$$6y^{2}-12-6y^{2}=-12y$$
$$-12=-12y$$
$$-12y=-12$$
$$y=\frac{-12}{-12}$$
$$y=1$$
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