I would like to show that $Var(aX + b) = a^2 Var(X)$ using a different proof from my book
Let $Y = aX + b$
$Var(Y) = E(Y^2) - E[Y]^2$
= $E[(aX + b)^2] - E(aX + b)^2$
= $E[(a^2 X^2 + b^2 + 2abX] - (a\mu+b)^2$
= $a^2E(X^2) + b^2 + 2abE[X] - (a^2\mu^2 + b^2 + 2ab\mu)$
= $a^2E(X^2) + b^2 + 2ab\mu - (a^2E[X]^2 + b^2 + 2ab\mu)$
= $a^2E[X^2] - a^2E[X]^2$
= $a^2(E[X^2]-E[X]^2)$
= $a^2 Var(X)$
My book does like three lines and they start out with $Var(aX + b) = E[(aX + b - (a\mu + b) ) ^2]$. I don't understand why they substract $(a\mu + b)$ instead of $\mu$ because $Var(X) = E[(X - \mu)^2]$
Is my proof correct?