It looks like you're asking to solve for the coefficients [tex]a,b,c[/tex] such that
[tex]2x^2 - 5 = a(x+2)^2 - b(x+1) + c[/tex]
Expanding the right side gives
[tex]2x^2 - 5 = ax^2 + 4ax + 4a - bx - b + c[/tex]
[tex]2x^2 - 5 = ax^2 + (4a - b)x + (4a - b + c)[/tex]
The coefficients on both sides must be equal, so
[tex]\begin{cases}a = 2 \\ 4a-b = 0 \\ 4a - b + c = -5\end{cases}[/tex]
Solving the system yields [tex]\boxed{a=2 \implies b = 8 \implies c = -5}[/tex].
