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The expression in the image asks to resolve the fraction \(\frac{x + 18}{(x - 3)(2x + 1)}\) into partial fractions.

### Step-by-Step Solution:

We are given:

\[
\frac{x + 18}{(x - 3)(2x + 1)}
\]

We want to express this as the sum of two simpler fractions:

\[
\frac{x + 18}{(x - 3)(2x + 1)} = \frac{A}{x - 3} + \frac{B}{2x + 1}
\]

To find \(A\) and \(B\), we follow these steps:

1. **Multiply both sides by \((x - 3)(2x + 1)\)** to eliminate the denominators:

\[
x + 18 = A(2x + 1) + B(x - 3)
\]

2. **Expand both sides**:

\[
x + 18 = A(2x + 1) + B(x - 3)
\]
\[
x + 18 = 2Ax + A + Bx - 3B
\]
\[
x + 18 = (2A + B)x + (A - 3B)
\]

3. **Compare coefficients of \(x\) and the constant term** on both sides:

For the \(x\)-terms:  
\[
1 = 2A + B
\]

For the constant terms:  
\[
18 = A - 3B
\]

Now, we have a system of equations:

\[
1. \quad 2A + B = 1
\]
\[
2. \quad A - 3B = 18
\]

4. **Solve the system of equations**:

From equation (1):  
\[
B = 1 - 2A
\]

Substitute this into equation (2):

\[
A - 3(1 - 2A) = 18
\]
\[
A - 3 + 6A = 18
\]
\[
7A - 3 = 18
\]
\[
7A = 21
\]
\[
A = 3
\]

Now substitute \(A = 3\) back into \(B = 1 - 2A\):

\[
B = 1 - 2(3) = 1 - 6 = -5
\]

5. **Write the partial fraction decomposition**:

Thus, the partial fraction decomposition of \(\frac{x + 18}{(x - 3)(2x + 1)}\) is:

\[
\frac{x + 18}{(x - 3)(2x + 1)} = \frac{3}{x - 3} + \frac{-5}{2x + 1}
\]

Or equivalently:

\[
\frac{x + 18}{(x - 3)(2x + 1)} = \frac{3}{x - 3} - \frac{5}{2x + 1}
\]

### Do you have any questions about the solution? Would you like more details on a specific step?

Here are some related questions you might want to explore:

1. What are partial fractions used for in calculus?
2. How do you handle repeated linear factors in partial fraction decomposition?
3. How would you decompose a fraction with a quadratic denominator?
4. What happens if the numerator has a higher degree than the denominator?
5. How can partial fractions be applied to solving integrals?

**Tip:** Always verify the partial fraction by adding the fractions back together to ensure it equals the original expression.

Resolve into partial fractions: \(\frac{x + 18}{(x - 3)(2x + 1)}\)

Learn how to decompose the rational expression \(\frac{x + 18}{(x - 3)(2x + 1)}\) into partial fractions with a detailed step-by-step solution. This problem covers key algebraic concepts and partial fraction decomposition. Suitable for students in Grades 10-12.

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