To solve $(9 - 4i)(5 + i)$, let's expand it using the distributive property, also known as the FOIL method:

1. First: $9 \times 5 = 45$
2. Outer: $9 \times i = 9i$
3. Inner: $-4i \times 5 = -20i$
4. Last: $-4i \times i = -4i^2$

Now we combine these terms:

$(9 - 4i)(5 + i) = 45 + 9i - 20i - 4i^2$

Next, simplify the expression:

• Combine the imaginary terms: $9i - 20i = -11i$.
• Recall that $i^2 = -1$, so $-4i^2 = -4 \times -1 = 4$.

Now the expression becomes:

$45 - 11i + 4$

Combine the real terms:

$49 - 11i$

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

$49 - 11i$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. How would the solution change if both terms had imaginary parts, like $(3 + 2i)(2 + 5i)$?
2. What is the geometric interpretation of multiplying complex numbers?
3. How does the distributive property apply to complex numbers?
4. What happens if we square a complex number, such as $(2 + 3i)^2$?
5. How can we verify our solution using the modulus and argument of complex numbers?

Tip: When multiplying complex numbers, remember that $i^2 = -1$, which simplifies expressions involving powers of $i$.