I would like to simplify complicated expressions by defining some variable that is a combination of other variables that appear in the expression, but without eliminating the original variables.
As a simple example, I would like to convert
a^2 + a*b^2 + b
a*c + b
by introducing the relation
c == a + b^2
I saw this question, but the answers require that one variable be eliminated. The best expressions I could get with those strategies are
b - b^2 c + c^2 a*c - Sqrt[-a + c] a*c + Sqrt[-a + c]
depending on whether I eliminate
b, but these are not meaningfully simpler than the original expression. Is there any way to introduce a new variable without eliminating one of the dependent variables in the expression? If it matters, I have Mathematica 10.3.