To solve $(3 + 8i)^2$, we'll use the formula for squaring a binomial, which is $(a + b)^2 = a^2 + 2ab + b^2$.

Given: $(3 + 8i)^2$

Let's break it down:

1. Square the real part: $3^2 = 9$

2. Square the imaginary part: $(8i)^2 = 64i^2$ Since $i^2 = -1$, this becomes: $64i^2 = 64(-1) = -64$

3. Multiply the real and imaginary parts and double the product: $2 \times 3 \times 8i = 48i$

Now, sum all these components together: $9 - 64 + 48i = -55 + 48i$

So, $(3 + 8i)^2 = -55 + 48i$.

Would you like any additional details or explanations?

Here are some related questions:

1. What is $(4 - 7i)^2$?
2. How do you multiply two different complex numbers, like $(3 + 8i)(2 - 5i)$?
3. What is the geometric interpretation of multiplying a complex number by itself?
4. How can you use the distributive property to multiply complex numbers?
5. What is the significance of the imaginary unit $i$ in complex numbers?

Tip: When multiplying complex numbers, remember to apply the distributive property and the identity $i^2 = -1$ carefully to simplify the expression.