\frac{4x^3-3x^2+22x-7}{\left(x^2+4\right)^2} partial fraction

To decompose the rational function

\[
\frac{4x^3 - 3x^2 + 22x - 7}{(x^2 + 4)^2}
\]

into partial fractions, we recognize that the denominator is the square of a quadratic, which means we need to account for both \((x^2 + 4)\) and \((x^2 + 4)^2\) in our decomposition. The form of the partial fraction decomposition would be:

\[
\frac{4x^3 - 3x^2 + 22x - 7}{(x^2 + 4)^2} = \frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2}
\]

### Steps:

1. **Multiply both sides by** \((x^2 + 4)^2\) to clear the denominator:
   
   \[
   4x^3 - 3x^2 + 22x - 7 = (Ax + B)(x^2 + 4) + (Cx + D)
   \]

2. **Expand the right-hand side**:
   \[
   (Ax + B)(x^2 + 4) = Ax^3 + 4Ax + Bx^2 + 4B
   \]
   So,
   \[
   (Ax + B)(x^2 + 4) + (Cx + D) = Ax^3 + Bx^2 + 4Ax + 4B + Cx + D
   \]
   Combine like terms:
   \[
   Ax^3 + Bx^2 + (4A + C)x + (4B + D)
   \]

3. **Set up a system of equations** by comparing coefficients of powers of \(x\) on both sides:
   
   - \(x^3\) term: \(A = 4\)
   - \(x^2\) term: \(B = -3\)
   - \(x^1\) term: \(4A + C = 22\)
   - Constant term: \(4B + D = -7\)

4. **Solve the system**:
   - From \(A = 4\), substitute into the equation \(4A + C = 22\):
     \[
     4(4) + C = 22 \implies C = 6
     \]
   - From \(B = -3\), substitute into the equation \(4B + D = -7\):
     \[
     4(-3) + D = -7 \implies D = 5
     \]

Thus, the partial fraction decomposition is:

\[
\frac{4x^3 - 3x^2 + 22x - 7}{(x^2 + 4)^2} = \frac{4x - 3}{x^2 + 4} + \frac{6x + 5}{(x^2 + 4)^2}
\]

---

Would you like more details or explanations on any part of this? 

Here are five questions to deepen your understanding:
1. How do you determine the structure of the partial fraction decomposition for a quadratic factor?
2. Why is it necessary to include both \((x^2 + 4)\) and \((x^2 + 4)^2\) in the decomposition?
3. Can this decomposition process be applied to higher powers of irreducible quadratics?
4. What would change if the quadratic had complex roots instead of being irreducible over the reals?
5. How does this method apply to other types of rational functions?

**Tip**: Always check that the degree of the numerator is less than that of the denominator before starting partial fraction decomposition. If not, use polynomial long division first.

\frac{4x^3-3x^2+22x-7}{(x^2+4)^2} partial fraction

This guide walks through the partial fraction decomposition of \frac{4x^3 - 3x^2 + 22x - 7}{(x^2 + 4)^2}. Learn step-by-step how to decompose this rational function, identify key coefficients, and solve using algebraic methods. Suitable for Grades 11-12 and early university students.

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