To solve the expression \(\frac{76 - 72i}{11 + 4i}\), we'll use the method of multiplying both the numerator and the denominator by the conjugate of the denominator.

### Steps:

1. **Identify the conjugate of the denominator:**
   \[
   \text{Conjugate of } (11 + 4i) = (11 - 4i)
   \]

2. **Multiply the numerator and denominator by this conjugate:**
   \[
   \frac{76 - 72i}{11 + 4i} \times \frac{11 - 4i}{11 - 4i} = \frac{(76 - 72i)(11 - 4i)}{(11 + 4i)(11 - 4i)}
   \]

3. **Expand the numerator:**
   \[
   (76 - 72i)(11 - 4i) = 76 \times 11 - 76 \times 4i - 72i \times 11 + 72i \times 4i
   \]
   \[
   = 836 - 304i - 792i + 288i^2
   \]
   \[
   \text{Since } i^2 = -1, \text{ substitute it:}
   \]
   \[
   = 836 - 1096i + 288(-1)
   \]
   \[
   = 836 - 1096i - 288
   \]
   \[
   = 548 - 1096i
   \]

4. **Expand the denominator:**
   \[
   (11 + 4i)(11 - 4i) = 11^2 - (4i)^2
   \]
   \[
   = 121 - 16i^2
   \]
   \[
   \text{Since } i^2 = -1:
   \]
   \[
   = 121 - 16(-1)
   \]
   \[
   = 121 + 16 = 137
   \]

5. **Final simplified form:**
   \[
   \frac{548 - 1096i}{137} = \frac{548}{137} - \frac{1096i}{137}
   \]
   \[
   = 4 - 8i
   \]

So, \(\frac{76 - 72i}{11 + 4i} = 4 - 8i\).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the conjugate of a complex number?
2. Can you explain why multiplying by the conjugate helps in simplifying complex fractions?
3. What are other methods to simplify complex number expressions?
4. How does one perform division with complex numbers geometrically?
5. Can you apply this method to other complex number divisions?

### Tip:
When dividing complex numbers, always multiply by the conjugate of the denominator to eliminate the imaginary part from the denominator.

Learn how to solve the complex number division problem (76-72i)/(11+4i) using the conjugate of the denominator. Detailed step-by-step solution provided with explanations on complex numbers and their division. Suitable for advanced high school students and beyond.

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