Coefficient of $x^{n-1}$ in the given expansion

Question

The problem I am facing is that with each term, number of ways to achieve $x^{n-1}$ is increasing, so it is getting very difficult to club all the cases together. Please provide some insight.

Answer 1

Let $a=x+3$ and $b=x+2$. Note that $$a^{n+1}-b^{n+1}=(a-b)(a^n +a^{n-1}b+\cdots +b^n).$$ In our case we have $a-b=1$. So we want the coefficient of $x^{n-1}$ in the expansions of $(x+3)^{n+1}$ and $(x+2)^{n+1}$. These are not hard to compute. The first is $3^2\binom{n+1}{2}$ and the second is $2^2\binom{n+1}{2}$. Subtract.

Answer 2

Observe that

$$A^{n+1}-B^{n+1}=(A-B)(A^n+A^{n-1}B+\ldots+AB^{n-1}+B^n)$$

and now put $\;A=x+3\,,\,\,B=x+2\;$