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To express \(\frac{x + 3}{(x - 2)(x + 4)}\) as partial fractions, we proceed as follows:

1. **Set up the partial fraction decomposition:**
   Since the denominator has two distinct linear factors, \(x - 2\) and \(x + 4\), we can write:
   \[
   \frac{x + 3}{(x - 2)(x + 4)} = \frac{A}{x - 2} + \frac{B}{x + 4}
   \]
   where \(A\) and \(B\) are constants that we need to determine.

2. **Multiply through by the denominator** \((x - 2)(x + 4)\) to clear the fractions:
   \[
   x + 3 = A(x + 4) + B(x - 2)
   \]

3. **Expand and collect terms:**
   Expanding both sides, we get:
   \[
   x + 3 = Ax + 4A + Bx - 2B
   \]
   Grouping the terms by powers of \(x\), this becomes:
   \[
   x + 3 = (A + B)x + (4A - 2B)
   \]

4. **Set up equations by equating coefficients:**
   For the coefficients of \(x\) and the constant terms, we have:
   - For \(x\): \(A + B = 1\)
   - For the constant term: \(4A - 2B = 3\)

5. **Solve the system of equations:**
   From \(A + B = 1\), we get \(B = 1 - A\).

   Substitute \(B = 1 - A\) into the second equation:
   \[
   4A - 2(1 - A) = 3
   \]
   Simplify:
   \[
   4A - 2 + 2A = 3
   \]
   \[
   6A = 5
   \]
   \[
   A = \frac{5}{6}
   \]

   Substitute \(A = \frac{5}{6}\) into \(B = 1 - A\):
   \[
   B = 1 - \frac{5}{6} = \frac{1}{6}
   \]

6. **Write the final partial fraction decomposition:**
   \[
   \frac{x + 3}{(x - 2)(x + 4)} = \frac{5}{6(x - 2)} + \frac{1}{6(x + 4)}
   \]

Would you like further details on any step or have any questions?

### Follow-up Questions
1. What are partial fractions used for in calculus?
2. How would you handle partial fraction decomposition if the denominator had a repeated factor?
3. Can partial fraction decomposition be applied to a denominator with irreducible quadratic terms?
4. How would you solve for \(A\) and \(B\) using a different method, like substituting specific values for \(x\)?
5. How does partial fraction decomposition help in integrating rational functions?

**Tip:** Always check that the degree of the numerator is lower than the degree of the denominator before attempting partial fraction decomposition.

Express \( \frac{x + 3}{(x - 2)(x + 4)} \) as partial fractions.

Learn how to express the rational function \( \frac{x + 3}{(x - 2)(x + 4)} \) as partial fractions. This solution explains the decomposition process and involves solving a system of equations to determine the constants. Suitable for students in Grades 10-12.

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