Firstly please check your input (or copy mine):
q1 = (-2 c + 2 a \[Alpha] - a b \[Alpha] +
b c \[Alpha]^2)/(4 \[Alpha] - b^2 \[Alpha]);
q2 = (b c + 2 a \[Alpha] - a b \[Alpha] -
2 c \[Alpha]^2)/(4 \[Alpha] - b^2 \[Alpha]) ;
cs = (q1^2 + q2^2 + 2 b q1 q2)/2
Out:
$$\frac{1}{2} \left(\frac{\left(2 a \alpha -a \alpha b+\alpha ^2 b c-2 c\right)^2}{\left(4 \alpha -\alpha b^2\right)^2}+\frac{2 b \left(2 a \alpha -a \alpha b+b c-2 \alpha ^2 c\right) \left(2 a \alpha -a \alpha b+\alpha ^2 b c-2 c\right)}{\left(4 \alpha -\alpha b^2\right)^2}+\frac{\left(2 a \alpha -a \alpha b+b c-2 \alpha ^2 c\right)^2}{\left(4 \alpha -\alpha b^2\right)^2}\right)$$
Secondly it's not clear why form
$\alpha \left(4 - b^2\right)^2$ not suitable?
