https://file.aiquickdraw.com/mathbot/file/17299396266820553e8te.jpg

To solve the problem, we are given the expression:

\[
(x - (-5 + 6i))(x - (-5 - 6i))
\]

### Step-by-Step Solution
1. **Rewrite the Expression**:
   The expression simplifies to:
   \[
   (x + 5 - 6i)(x + 5 + 6i)
   \]

2. **Recognize the Conjugate Product Pattern**:
   This expression is in the form \((a - bi)(a + bi)\), which is a difference of squares. Using the identity \((a - bi)(a + bi) = a^2 + b^2\), we let:
   - \(a = x + 5\)
   - \(b = 6\)

3. **Apply the Formula**:
   \[
   (x + 5)^2 + (6)^2
   \]

4. **Expand and Simplify**:
   - First, expand \((x + 5)^2\):
     \[
     (x + 5)^2 = x^2 + 10x + 25
     \]
   - Next, calculate \(6^2\):
     \[
     6^2 = 36
     \]

5. **Combine Terms**:
   \[
   x^2 + 10x + 25 + 36 = x^2 + 10x + 61
   \]

### Final Answer
\[
x^2 + 10x + 61
\]

Would you like a more detailed explanation on any step or have additional questions?

---

Here are five related questions to deepen your understanding:
1. What is the general formula for multiplying complex conjugates?
2. How do you expand a binomial expression like \((x + 5)^2\)?
3. Why does multiplying conjugates eliminate the imaginary part?
4. How can this process be applied to other expressions involving complex numbers?
5. What is the significance of the result being a polynomial?

**Tip:** When multiplying conjugates, remember that the imaginary parts cancel out, leaving only a real number result.

Multiply the expressions involving complex conjugates: \[(x - (-5 + 6i))(x - (-5 - 6i))\]

Solve and simplify the multiplication of expressions involving complex conjugates \[(x - (-5 + 6i))(x - (-5 - 6i))\]. Learn to apply the difference of squares formula to simplify expressions with complex numbers. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation