Question

multiply and simplify the following complex numbers: \\((-1 + 4i) \cdot (4 - 3i)\\)

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
\[

$$\begin{align*} (-1 + 4i) \cdot (4 - 3i)&= -1 \cdot 4 + (-1) \cdot (-3i) + 4i \cdot 4 + 4i \cdot (-3i)\\ &= -4 + 3i + 16i - 12i^2 \end{align*}$$

\]

Step2: Simplify using \(i^2 = -1\)

Recall that \(i^2 = -1\), so substitute \(-1\) for \(i^2\) in the expression:
\[

$$\begin{align*} -4 + 3i + 16i - 12i^2&= -4 + 3i + 16i - 12(-1)\\ &= -4 + 3i + 16i + 12 \end{align*}$$

\]

Step3: Combine like terms

Combine the real parts and the imaginary parts separately:
\[

$$\begin{align*} -4 + 12 + (3i + 16i)&= 8 + 19i \end{align*}$$

\]

Answer:

\(8 + 19i\)