We multiply throughout by (x - 1)³(x + 1):

2x² - 3 = A(x - 1)²(x + 1) + B(x - 1)(x + 1) + C(x + 1) + D(x - 1)³

We do not need to expand all this out. Instead, we use appropriate substitutions to find the values of A to D.

Let x = 1:

LHS = -1

RHS = 2C

So C = -1/2

Let x = -1:

LHS = -1

RHS = -8D

So D = 1/8

We now compare the coefficients of x³ on both sides and then compare the constant values on both sides.

Coefficient of x³ on LHS = 0

Coefficient of x³ on RHS = A + D

But since D = 1/8, we have A = -1/8.

Constant term on LHS = -3

Constant term on RHS = A - B + C - D

But since we know 3 values now, we have: B = 9/4.

So