The given expression is $(12 - 9i)^2$, and you need to simplify it into the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

To simplify:

1. Use the formula $(x - y)^2 = x^2 - 2xy + y^2$.

   • Here, $x = 12$ and $y = 9i$.

2. Expand $(12 - 9i)^2$:

$(12 - 9i)^2 = 12^2 - 2(12)(9i) + (9i)^2$

Now, simplify each term:

• $12^2 = 144$,
• $-2(12)(9i) = -216i$,
• $(9i)^2 = 81i^2 = 81(-1) = -81$ (since $i^2 = -1$).

3. Combine these results:

$(12 - 9i)^2 = 144 - 216i - 81$

4. Simplify further:

$144 - 81 = 63$

So the simplified expression is:

$63 - 216i$

Thus, the expression in $a + bi$ form is:

$63 - 216i$

Would you like further explanation or details on this process?

Here are 5 related questions to expand your understanding:

1. How do you square a complex number using binomial expansion?
2. What is the significance of the imaginary unit $i$ in complex numbers?
3. How does $i^2 = -1$ affect calculations involving complex numbers?
4. Can you express the product of two complex numbers geometrically?
5. What are the real and imaginary components of a complex number?

Tip: When squaring complex numbers, always remember that $i^2 = -1$, which simplifies the calculation.