How to simplify this fraction with different powers? I happen to be stuck trying to simplify this:
$$\left[\frac{(3x+2)(x+1)^\frac{3}{2}-(\frac{3}{2}x^2+2x)(\frac{3}{2})(x+1)^\frac{1}{2}}{(x+1)^3}\right]$$
here's the simplified solution that I'm trying to figure out how it was reached

$$\frac{(3x+2)(x+1)^\frac{3}{2}-(\frac{3}{2}x^2+2x)(\frac{3}{2})(x+1)^\frac{1}{2}}{(x+1)^3}=$$
take the common factor $(x+1)^\frac{1}{2}$
$$=\frac{[(3x+2)(x+1)-(\frac{3}{2}x^2+2x)(\frac{3}{2})]\color{red}{(x+1)^\frac{1}{2}}}{(x+1)^3}=$$
$$=\frac{(3x+2)(x+1)-(\frac{3}{2}x^2+2x)(\frac{3}{2})}{(x+1)^3\color{red}{(x+1)^{-\frac{1}{2}}}}=\frac{(3/4)x^2+2x+2}{(x+1)^{\frac{5}{2}}}$$
