Question

multiply and simplify the following complex numbers: \\((-4 - 5i) \cdot (1 - i)\\)

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first complex number by each term in the second complex number:
$$(-4 - 5i)\cdot(1 - i)=-4\cdot1+(-4)\cdot(-i)+(-5i)\cdot1+(-5i)\cdot(-i)$$

Step2: Simplify each product

Simplify each term:
$$-4\cdot1 = -4$$
$$(-4)\cdot(-i)=4i$$
$$(-5i)\cdot1=-5i$$
$$(-5i)\cdot(-i)=5i^{2}$$
So the expression becomes:
$$-4 + 4i-5i + 5i^{2}$$

Step3: Combine like terms and use \(i^{2}=-1\)

Combine the \(i\) terms: \(4i-5i=-i\)
Substitute \(i^{2}=-1\) into \(5i^{2}\): \(5i^{2}=5\times(-1)=-5\)
Now the expression is:
$$-4 - i-5$$
Combine the constant terms: \(-4-5=-9\)
So we have:
$$-9 - i$$

Answer:

\(-9 - i\)