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To find the partial fractions of the given rational function, we need to follow these steps:

Given rational function:

\[
\frac{4x^2 + 13x - 9}{x^3 + 2x^2 - 3x}
\]

First, factor the denominator:

\[
x^3 + 2x^2 - 3x = x(x^2 + 2x - 3) = x(x + 3)(x - 1)
\]

So, we rewrite the fraction as:

\[
\frac{4x^2 + 13x - 9}{x(x + 3)(x - 1)}
\]

Next, we decompose it into partial fractions. The general form will be:

\[
\frac{4x^2 + 13x - 9}{x(x + 3)(x - 1)} = \frac{A}{x} + \frac{B}{x + 3} + \frac{C}{x - 1}
\]

Multiply through by the common denominator \(x(x + 3)(x - 1)\) to clear the fractions:

\[
4x^2 + 13x - 9 = A(x + 3)(x - 1) + Bx(x - 1) + Cx(x + 3)
\]

Now, we need to find the values of \(A\), \(B\), and \(C\). Expand and simplify the right-hand side:

\[
4x^2 + 13x - 9 = A(x^2 - x + 3x - 3) + B(x^2 - x) + C(x^2 + 3x)
\]

\[
= A(x^2 + 2x - 3) + B(x^2 - x) + C(x^2 + 3x)
\]

Combine like terms:

\[
= (A + B + C)x^2 + (2A - B + 3C)x - 3A
\]

Set the coefficients equal to those on the left-hand side:

\[
4x^2 + 13x - 9 = (A + B + C)x^2 + (2A - B + 3C)x - 3A
\]

This gives us the system of equations:

1. \(A + B + C = 4\)
2. \(2A - B + 3C = 13\)
3. \(-3A = -9\)

From the third equation:

\[
A = 3
\]

Substitute \(A = 3\) into the first and second equations:

1. \(3 + B + C = 4 \implies B + C = 1\)
2. \(2(3) - B + 3C = 13 \implies 6 - B + 3C = 13 \implies -B + 3C = 7\)

Now solve the system:

1. \(B + C = 1\)
2. \(-B + 3C = 7\)

Add the two equations:

\[
B + C - B + 3C = 1 + 7 \implies 4C = 8 \implies C = 2
\]

Substitute \(C = 2\) back into \(B + C = 1\):

\[
B + 2 = 1 \implies B = -1
\]

So, the values are:

\[
A = 3, B = -1, C = 2
\]

Thus, the partial fraction decomposition is:

\[
\frac{4x^2 + 13x - 9}{x(x + 3)(x - 1)} = \frac{3}{x} + \frac{-1}{x + 3} + \frac{2}{x - 1}
\]

Would you like further details or have any questions about this process? 

Here are some additional questions you can ask:
1. How do you solve a system of linear equations?
2. Can you explain more about partial fraction decomposition?
3. How do you factor complex polynomials?
4. What are some common techniques for integrating rational functions?
5. How do partial fractions apply to solving differential equations?

**Tip:** When performing partial fraction decomposition, always ensure the degree of the numerator is less than the degree of the denominator. If it's not, perform polynomial division first.

Learn how to decompose the rational function 4x^2 + 13x - 9 / (x^3 + 2x^2 - 3x) into partial fractions step-by-step. This problem covers partial fraction decomposition, polynomial factorization, and solving systems of linear equations. Suitable for advanced high school students.

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