We need to perform the multiplication $(9 + 7i)(5 + 2i)$ and simplify the result.

Step 1: Apply the distributive property (FOIL method)

We distribute each term in the first binomial to each term in the second binomial:

$(9 + 7i)(5 + 2i) = 9(5) + 9(2i) + 7i(5) + 7i(2i)$

This simplifies to:

$= 45 + 18i + 35i + 14i^2$

Step 2: Simplify the result

• Combine the real parts: $45$
• Combine the imaginary parts: $18i + 35i = 53i$
• Simplify $14i^2$. Since $i^2 = -1$, we have: $14i^2 = 14(-1) = -14$

Now, put it all together:

$45 + 53i - 14 = (45 - 14) + 53i$

This simplifies to:

$31 + 53i$

Final Answer:

$31 + 53i$

Would you like more details on this process, or do you have any other questions?

Further Questions:

1. What is the general rule for multiplying two complex numbers?
2. How do we handle complex conjugates in multiplication?
3. Can we factorize the expression further?
4. How do we add or subtract complex numbers?
5. What is the geometric interpretation of multiplying complex numbers?

Tip: When multiplying complex numbers, always remember to use the fact that $i^2 = -1$, which can help