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}`.