The partial fraction expansion will look like
(x - 2)/((x ² + 1) (x - 1)²) = (ax + b)/(x ² + 1) + c/(x - 1) + d/(x - 1)²
Get everything in terms of a common denominator, and compare the numerators on both sides:
x - 2 = (ax + b) (x - 1)² + c (x ² + 1) (x - 1) + d (x ² + 1)
Expand the right side:
x - 2 = (ax + b) (x - 1)² + c (x ² + 1) (x - 1) + d (x ² + 1)
x - 2 = (a + c) x ³ + (-2a + b - c + d) x ² + (a - 2b + c) x + b - c + d
Match up the coefficients and solve the resulting system of equations:
a + c = 0
-2a + b - c + d = 0
a - 2b + c = 1
b - c + d = -2
==> a = -1, b = -1/2, c = 1, d = -1/2
So the expansion into partial fractions is
(x - 2)/((x ² + 1) (x - 1)²) = (-x - 1/2)/(x ² + 1) + 1/(x - 1) - 1/(2 (x - 1)²)
… = -(2x + 1)/(2 (x ² + 1)) + 1/(x - 1) - 1/(2 (x - 1)²)