HELP WITH PARTIAL DECOMPOSITION! FULL POINTS FOR THE BEST ANSWER!

Write the terms for the partial decomposition. Do not solve for constants.

x^3+2x^2-3 / (x^2+2)(x^2-2x+1)

Thank you!

Note that x^2 - 2x + 1 can be factored further to (x - 1)^2, so:
(x^3 + 2x^2 - 3)/[(x^2 + 2)(x^2 - 2x + 1)] = (x^3 + 2x^2 - 3)/[(x^2 + 2)(x - 1)^2].

By Partial Fractions, we see that for some A, B, C, and D:
(x^3 + 2x^2 - 3)/[(x^2 + 2)(x - 1)^2] = (Ax + B)/(x^2 + 2) + C/(x - 1) + D/(x - 1)^2.

Note that the quadratic denominator gets a linear numerator and that (x - 1)^2 is split up.

This is the general form of the partial fraction expansion.

I hope this helps!

(x^3 + 2x² - 3)/(x² + 2)(x² - 2x + 1) = (Ax + B)/(x² + 2) + (Cx + D)/(x - 1)²