Chain rule for $ y = (x^2 + x^3)^4$ I am trying to find the derivative of $ y = (x^2 + x^3)^4$
and it seems pretty simple I get
$4(x^2+x^3)^3 (2x+3x^2)$ This seems to be the proper answer to me but the book gets 
$4x^7 (x+1)^3 (3x+2)$ and I have no idea how that happened, what process the author went through or why. My answer seems to be a better and more accurate answer since that is what the chain rule will give you. 
Why is my answer wrong?

You did the differentiation correctly. The book's solution is the same as your answer, but in simplified form. 
To obtain the book's answer from yours: factor $x^2$ from the $(x^2+x^3)$ term in your expression and apply the rule $(ab)^n=a^nb^n$, factor $x$ from the $(2x+3x^2)$ term, and finally combine the $x^6$ and $x$ terms.
$$\eqalign{
4(\color{maroon}{x^2+x^3})^3 (\color{darkgreen}{2x+3x^2})
&= 4  \bigl(\color{maroon}{ x^2(1+x)}\bigr)^3\cdot\color{darkgreen}{ x(2+3x)}\cr 
&=4\cdot (x^2)^3(1+x)^3\cdot x(2+3x)\cr
&=4x^6(1+x)^3\cdot x(2+3x)\cr
&=4x^7(x+1)^3(3x+2).
}
$$
