When I have a form I wish to keep preserved, I might replace it by its own variable — though it makes doing algebra with it difficult — or do the following, which allows for algebra to be carried out:
denom = Denominator[q1]
Simplify[denom^2*cs]/denom^2
The first way, if you need to do some algebra, is not a way I'd recommend. It's for when I don't need any algebraic simplification. However, it might be done like this:
Clear[\[Alpha]b];
(*\[Alpha]b/:Power[\[Alpha]b,p_?Positive]:=Echo@Power[4 \[Alpha]-b^2 \[Alpha],p];*)
\[Alpha]b /: Plus[\[Alpha]b, c_] := 4 \[Alpha] - b^2 \[Alpha] + c;
\[Alpha]b /: Times[\[Alpha]b, c_] := c (4 \[Alpha] - b^2 \[Alpha]);
\[Alpha]bRule = \[Alpha]b -> 4 \[Alpha] - b^2 \[Alpha];
cs /. Reverse@\[Alpha]bRule // Simplify // ReplaceAll[\[Alpha]bRule]
You can't use the Power definition, because it will get invoked when the fractions are combined by Simplify. The denominator gets expanded, and you end up the same factored denominator that you were trying to avoid. You cannot control or always predict what will go on internally, which is why I don't recommend the approach.
Also the success of the first step, cs /. Reverse@\[Alpha]bRule, relies on the denominator in each term having a form that matches the pattern part of the rule. That happens to be the case here. We were lucky.
