To multiply the complex numbers $(10 - 3i)$ and $(4 - 6i)$, we can apply the distributive property (also known as the FOIL method for binomials). Here's the step-by-step calculation:

$(10 - 3i)(4 - 6i) = 10 \cdot 4 + 10 \cdot (-6i) + (-3i) \cdot 4 + (-3i) \cdot (-6i)$

Now calculate each term:

• $10 \cdot 4 = 40$
• $10 \cdot (-6i) = -60i$
• $(-3i) \cdot 4 = -12i$
• $(-3i) \cdot (-6i) = 18i^2$

Since $i^2 = -1$, the term $18i^2$ becomes $18(-1) = -18$.

Now, combine all the terms:

$40 - 60i - 12i - 18$

Simplify:

$40 - 18 = 22 \quad \text{and} \quad -60i - 12i = -72i$

So, the final result is:

$22 - 72i$

Answer: $(10 - 3i)(4 - 6i) = 22 - 72i$

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you simplify expressions involving complex conjugates?
2. What is the definition of the imaginary unit $i$?
3. How do you add or subtract complex numbers?
4. What is the polar form of a complex number?
5. Can complex numbers be multiplied using matrices?

Tip: When multiplying complex numbers, remember to apply the distributive property and simplify terms involving $i^2$ to real numbers!