Partial fraction

x³ + 9x² + 9x + 15
────────────
(x² + 1)(x² + 4x + 4)
=x³ + 9x² + 9x + 15
───────────
(x² + 1) (x + 2)²

Let the partial fraction be A .............. B................ C
─── + ───── + ────
x + 2........ (x + 2)²....... x² + 1
= A(x + 2)(x² + 1) + B(x² + 1) + C(x + 2)²
───────────────────────────
......................(x² + 1) (x + 2)²
= A(x³ + 2x² + x + 2) + B(x² + 1) + C(x² + 4x + 4)
───────────────────────────────
...................... (x² + 1) (x + 2)²
=Ax³ + (2A + B + C)x² + (A + 4C)x + 2A + B + 4C
────────────────────────────────
...................... (x² + 1) (x + 2)²

Comparing the coefficients of the numerator :A = 1
{
2A + B + C = 9
{
A + 4C = 9
{
2A + B + 4C = 15
⇒ A = 1 , B = 5 , C = 2
∴ The partial fraction
= 1 .............. 5................ 2
─── + ───── + ────
x + 2....... .(x + 2)²....... x² + 1