To solve \((4 - 3i)(-5 + 4i)\), we'll use the distributive property, also known as FOIL (First, Outer, Inner, Last) method for multiplying two binomials.

\[
(4 - 3i)(-5 + 4i) = 4(-5) + 4(4i) - 3i(-5) - 3i(4i)
\]

Now, let's calculate each part:

1. \(4 \times -5 = -20\)  (First)
2. \(4 \times 4i = 16i\)  (Outer)
3. \(-3i \times -5 = 15i\)  (Inner)
4. \(-3i \times 4i = -12i^2\)  (Last)

Next, combine like terms and simplify:

\[
-20 + 16i + 15i - 12i^2
\]

Combine the real and imaginary parts:

\[
-20 + 31i - 12i^2
\]

Since \(i^2 = -1\), substitute \(-1\) for \(i^2\):

\[
-20 + 31i - 12(-1) = -20 + 31i + 12
\]

Finally, combine the real parts:

\[
-20 + 12 + 31i = -8 + 31i
\]

So, the product is:

\[
\boxed{-8 + 31i}
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the magnitude of the complex number \(-8 + 31i\)?
2. How do you multiply complex numbers using the polar form?
3. What is the conjugate of the complex number \(-8 + 31i\)?
4. How can you represent complex numbers on an Argand diagram?
5. How do you find the real and imaginary parts of a complex number?

**Tip:** Remember that \(i^2 = -1\), which is crucial when simplifying products of complex numbers.

Explore how to multiply complex numbers (4 - 3i)(-5 + 4i) using the FOIL method and properties of imaginary units. Follow a detailed solution to simplify and find the product -8 + 31i. Suitable for high school students studying complex numbers.

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