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!