Let $a$ and $b$ be the coefficient of $x^3$ in $(1+x+2x^2+3x^3)^4$ and $(1+x+2x^2+3x^3+4x^4)^4$ respectively. Let $a$ and $b$ be the coefficient of $x^3$ in $(1+x+2x^2+3x^3)^4$ and $(1+x+2x^2+3x^3+4x^4)^4$ respectively.Find $(a-b).$

I tried to factorize $(1+x+2x^2+3x^3)$ and $(1+x+2x^2+3x^3+4x^4)$ into product of two binomials,but i could not.And i do not have any other method to solve it.

Here is a slightly different variation of the theme. If we introduce the polynomial
\begin{align*}
p(x)=1+x+2x^2+3x^3
\end{align*}
and write $[x^n]$ to denote the coefficient of $x^n$ of a polynomial,

we observe
  \begin{align*}
&[x^3]\left(p(x)\right)^4-[x^3]\left(p(x)+4x^4\right)^4\\
&\quad=[x^3]\left(p(x)\right)^4-[x^3]\left(p(x)\right)^4\tag{1}\\
&\quad=0
\end{align*}
  In (1) we use the fact that multiplication of $4x^4$ with $p(x)$ gives always terms with power of $x$ greater or equal to $4$ and so there is no contribution to $[x^3]$.

