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The given expression is \((8 + 9i)^2\), where \(i\) is the imaginary unit. We need to simplify this expression and write it in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

### Step-by-step solution:
1. Apply the binomial expansion formula: \((a + b)^2 = a^2 + 2ab + b^2\).
   - In this case, \(a = 8\) and \(b = 9i\).

2. Expand the expression:
   \[
   (8 + 9i)^2 = 8^2 + 2(8)(9i) + (9i)^2
   \]
   
3. Simplify each term:
   \[
   8^2 = 64
   \]
   \[
   2(8)(9i) = 144i
   \]
   \[
   (9i)^2 = 81i^2 = 81(-1) = -81
   \]

4. Combine the terms:
   \[
   (8 + 9i)^2 = 64 + 144i - 81
   \]
   \[
   = (64 - 81) + 144i
   \]
   \[
   = -17 + 144i
   \]

### Final answer:
\[
(8 + 9i)^2 = -17 + 144i
\]

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. How would you solve \((5 + 7i)^2\) using a similar approach?
2. What is the geometrical interpretation of squaring a complex number?
3. How does the expansion of \((a + b)^2\) change when both terms are complex?
4. Can you find the conjugate of the result \(-17 + 144i\)?
5. What happens if you square a purely imaginary number, like \((0 + 9i)^2\)?

**Tip**: Remember that \(i^2 = -1\), which is key to simplifying expressions involving imaginary numbers!

Learn how to simplify the complex number expression (8 + 9i)^2 step-by-step. This solution involves the binomial expansion and properties of imaginary numbers. Suitable for students in Grades 10-12.

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