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.

Let $p(x)=1+x+2x^2+3x^3$ and $q(x)=1+x+2x^2+3x^3+4x^4$. Then 
$$(q(x))^4-(p(x))^4=(q(x)-p(x))A(x)$$
for some polynomial $A(x)$. But $q(x)-p(x)=4x^4$, so $(q(x))^4-(p(x))^4$ is divisible by $x^4$. It follows that all coefficients of $x^k$ for $k\le 3$ are $0$.
