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The problem provided in the image is asking to solve for \( x \) and \( y \) in the equation:

\[
3 - 4i = (y + 4i)(x - yi)
\]

The problem also asks to ensure that any non-real solutions are expressed in the form \( a + bi \), where \( i = \sqrt{-1} \).

### Step-by-Step Solution

1. **Expand the Right-Hand Side**:
   \[
   (y + 4i)(x - yi) = yx - y^2i + 4ix - 4yi^2
   \]

2. **Simplify Using \( i^2 = -1 \)**:
   \[
   = yx - y^2i + 4ix + 4y
   \]
   \[
   = (yx + 4y) + (-y^2 + 4x)i
   \]

   So the equation becomes:
   \[
   3 - 4i = (yx + 4y) + (-y^2 + 4x)i
   \]

3. **Equating Real and Imaginary Parts**:
   - Real part: \( 3 = yx + 4y \)
   - Imaginary part: \( -4 = -y^2 + 4x \)

   We have the following system of equations:
   \[
   yx + 4y = 3 \quad \text{(Equation 1)}
   \]
   \[
   -y^2 + 4x = -4 \quad \text{(Equation 2)}
   \]

4. **Solve Equation 1 for \( x \)**:
   \[
   x(y + 4) = 3 \quad \Rightarrow \quad x = \frac{3}{y + 4}
   \]

5. **Substitute \( x \) into Equation 2**:
   \[
   -y^2 + 4\left(\frac{3}{y + 4}\right) = -4
   \]
   Multiply through by \( y + 4 \) to clear the fraction:
   \[
   -y^2(y + 4) + 12 = -4(y + 4)
   \]
   Simplify:
   \[
   -y^3 - 4y^2 + 12 = -4y - 16
   \]
   Rearrange all terms:
   \[
   y^3 + 4y^2 - 4y - 28 = 0
   \]

This is a cubic equation in \( y \). To solve this, you can try possible rational roots using the Rational Root Theorem or numerical methods.

Once you find \( y \), you can substitute it back into the equation \( x = \frac{3}{y + 4} \) to find \( x \).

### Next Steps

Would you like a detailed exploration of how to solve the cubic equation or further assistance with a specific part of the problem?

Here are some related questions you might find useful:

1. How do you solve cubic equations using the Rational Root Theorem?
2. Can all cubic equations be factored easily, and what techniques can be used?
3. What methods can you use to find the roots of a polynomial equation?
4. What is the significance of the imaginary unit \( i \) in complex numbers?
5. How do complex numbers help solve real-world problems?
6. How can you verify the solutions of a complex number equation?
7. What are the applications of solving equations with complex numbers in engineering?
8. How does the factorization of polynomials work when dealing with complex numbers?

**Tip:** Always verify your solutions by substituting them back into the original equation to ensure they satisfy the given equation.

Learn how to solve the complex number equation 3 - 4i = (y + 4i)(x - yi) step-by-step. This problem involves complex number operations and quadratic equation solving techniques. Suitable for high school students.

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