To multiply two complex numbers, you use the distributive property (FOIL method for binomials). Let's multiply the complex numbers $(-3 + 7i)$ and $(2 + 5i)$.

$(-3 + 7i) \cdot (2 + 5i)$

Step 1: Apply the distributive property (FOIL)

$(-3) \cdot 2 + (-3) \cdot 5i + (7i) \cdot 2 + (7i) \cdot 5i$

Step 2: Perform the multiplication

$= -6 - 15i + 14i + 35i^2$

Step 3: Simplify the result

We know that $i^2 = -1$, so:

$35i^2 = 35(-1) = -35$

Now, combine like terms:

$-6 + (-35) = -41$ $-15i + 14i = -i$

Final Answer:

$-41 - i$

So, the product of $(-3 + 7i)$ and $(2 + 5i)$ is $-41 - i$.

Would you like further clarification or details on any step?

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Here are 5 related questions to explore:

1. How do you divide two complex numbers?
2. What is the magnitude of a complex number?
3. What is the conjugate of a complex number, and how is it used in division?
4. How do you convert a complex number into polar form?
5. What is the real and imaginary part of a complex number?

Tip: When multiplying complex numbers, always remember to use $i^2 = -1$ to simplify terms!