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## Homework Statement

Hello!

Here is my second post on the subject partial fraction decomposition. The subject looks pretty easy to learn, but when I try exercises, I do not get to the correct answer. Please, take a look at the exercise below and help me to see my mistakes.

## Homework Equations

(-2x^2 + 20x - 68) / (x^3 + 4x^2 + 4x + 16)

## The Attempt at a Solution

Step 1: create a form for partial fraction decomposition by factoring the denominator:

x^3 + 4x^2 + 4x + 16 = (x^2 + 4) ( x + 4)

x^2 + 4 is in irreducible quadratic form, thus I will work with the above factors

Step 2: clear denominators

(-2x^2 + 20x - 68) (x^2 + 4) ( x + 4) / (x^3 + 4x^2 + 4x + 16) = A (x^2 + 4) ( x + 4) / (x + 4) + (Bx + C) (x^2 + 4) ( x + 4) / (x^2 + 4)

-2x^2 + 20x - 68 = x^2 (A + B) + x ( C + 4B) + 4C + 4A

Step 3: find values of A, B, C

A + C = -2

C + 4B = 20

4C + 4A = -68 (A + C) = -17

Matrix A

1 0 1

0 1 4

1 0 1

Matrix B

-2

20

-17

And the determinant of matrix A is zero, hence I can't solve the task using Cramer's rule. Obviously, there are mistakes in my approach. What are they? Please, help me to see them.

Thank you!

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