Simplifying a binomial sum

Question

I came across the following expression $$ -1+(1+a^5)x+4a^4bx^{h+1}+6a^3b^2x^{2h+1}+4a^2b^3x^{3h+1}+ab^4x^{4h+1}=0 $$ which I would like to simplify. I though of using a sum with the binomial coefficient, maybe something like $$ 1-x=\sum_{j=0}^4\binom{4}{j}a^{5-j}b^jx^{jh+1} $$ but I'm not sure if I can simplify it further. Any ideas?

Answer 1

If you expand $(a+bx^h)^4$ you get

$$a^4 + 4a^3bx^h + 6 a^2b^2x^{2h} + 4ab^3x^{3h} + b^4x^{4h}.$$

So your equation is

$$-1 + x+ ax(a+bx^h)^4 = 0.$$

Answer 2

I would try something named »coefficient comparison«. For example, for $jh + 1 = 0$, we get $1 = \dots x^{jh + 1}$