Question

Given that 2x^2-5 = a(x+2)^2 _ b(x+1)+x for all values of X, find the value of a, of b and of c

Answer 1

It looks like you're asking to solve for the coefficients $a,b,c$ such that

$2x^2 - 5 = a(x+2)^2 - b(x+1) + c$

Expanding the right side gives

$2x^2 - 5 = ax^2 + 4ax + 4a - bx - b + c$

$2x^2 - 5 = ax^2 + (4a - b)x + (4a - b + c)$

The coefficients on both sides must be equal, so

$\begin{cases}a = 2 \\ 4a-b = 0 \\ 4a - b + c = -5\end{cases}$

Solving the system yields $\boxed{a=2 \implies b = 8 \implies c = -5}$.