This polynomial is divisible by $(x-2)$ so the remainder is $0$. How do we evaluate the product $ab$? $$P(x) = x^{2a+b-1} +x^{a-2b+5}-2x^{a+b-1}$$
This polynomial is divisible by $(x-2)$ so the remainder is $0$. How do we evaluate the product $ab$?
My attempt: 

This polynomial is divisible by $(x-2)$ so the remainder is $0$

$$P(x) = (x-2)Q(x) + 0 \implies P(2) = 0$$
Plugging $x = 2$
$$P(2) = (2)^{2a+b-1}+(2)^{a-2b+5}-2(2)^{a+b-1}$$
Let $2^a = u$ and $2^b = m$
$$0 = \dfrac{1}{2}(u^2m)+32um^{-2}-2\dfrac{1}{2}(um) = \dfrac{1}{2}(u^2m)+32um^{-2}-um$$
Multiplying both sides by $2m^2$
$$0 = u^2m^3+64u-2um^3$$
Factoring $u$
$$0 = u\biggr(um^3+64-2m^3\biggr ) $$
Here we get one solution for $u = 0$. But I'm not sure If I went correctly. 

We continue from the last line, since it has been pointed out in comments that $u\ne0$:
$$0=um^3+64-2m^3=(u-2)m^3+64$$
$$64=(2-u)m^3$$
Given that $u,m$ are powers of 2, $2-u$ is positive, but $u>0$ so $u=1$ and $m=4$, i.e. $a=0,b=2$ and $ab=0$.
