2z² + z + C = A (z + 1)² + B (z + 1) + 4
Expand the right side completely and collect terms with the same power of z :
2z² + z + C = A (z² + 2z + 1) + B (z + 1) + 4
2z² + z + C = Az² + 2Az + A + Bz + B + 4
2z² + z + C = Az² + (2A + B)z + A + B + 4
Now match up terms on either side that have the same power of z. This gives
2 = A ................. (2nd power)
1 = 2A + B ......... (1st power)
C = A + B + 4 .... (0th power)
Solve for A, B, and C. The first equation gives A = 2 right away, so in the second equation we get
1 = 2*2 + B
B = -3
and in the third equation we get
C = 2 + (-3) + 4
C = 3
