x +2 = A(x -3) +B(x +1) ------(i)

x +2 = Ax -3A +Bx +B

x +2 = (A +B)x +(-3A +B) ------(ii)

For (ii) to be true,

a)the coefficient of x in the LHS must be equal to the coefficient of x in the RHS.

Or, 1 = A +B -----------(iii)

b) and the constants on both sides of the equation (ii) must also be equal.

Or,

2 = -3A +B ------------(iv)

Now solve (iii) and (iv) simultaneously to find A and B.

You'd get

A = -1/4

B = 5/4