Question

express in simplest ( a + bi ) form: ((1 - 2xi)(1 + 4i) - (1 - 2xi)(1 + 6i))

Explanation:

Step1: Factor out the common term

Notice that both terms have a common factor of \((1 - 2xi)\), so we can factor it out:
\[
(1 - 2xi)[(1 + 4i) - (1 + 6i)]
\]

Step2: Simplify the expression inside the brackets

Simplify \((1 + 4i) - (1 + 6i)\):
\[
(1 + 4i) - (1 + 6i)=1 + 4i - 1 - 6i=(1 - 1)+(4i - 6i)= - 2i
\]

Step3: Multiply the factored term with the simplified bracket

Now we have \((1 - 2xi)(-2i)\). Distribute \(-2i\) across the terms in the parentheses:
\[
(1 - 2xi)(-2i)=1\times(-2i)-2xi\times(-2i)=-2i + 4xi^{2}
\]

Step4: Substitute \(i^{2}=-1\)

Since \(i^{2}=-1\), substitute this into the expression:
\[
-2i + 4xi^{2}=-2i + 4x(-1)=-4x-2i
\]

Answer:

\(-4x - 2i\)