College Algebra, Chapter 1, 1.4, Section 1.4, Problem 46

Evaluate the expression $\displaystyle \frac{(1+2i)(3-i)}{2+i}$ in the form of $a + bi$.


$\begin{equation} \begin{aligned} &= \frac{(1+2i)(3-i)}{2+i}\\ \\ &= \frac{3-i+6i-2i^2}{2+i} && \text{Use FOIL method}\\ \\ &= \frac{3+5i -2 (-1)}{2+i} && \text{recall that } i^2 = -1\\ \\ &= \frac{5+5i}{2+i} && \text{Simplify the numerator}\\ \\ &= \left( \frac{5+5i}{2+i} \right) \left( \frac{2-i}{2-i} \right) && \text{Multiply by the denominator of the conjugate}\\ \\ &= \frac{10-5i+10i-5i^2}{2^2 - i^2} && \text{Apply FOIL method}\\ \\ &= \frac{10-5i+10i-5(-1)}{4-(-1)} && \text{recall that } i^2 = -1\\ \\ &= \frac{15 + 5i}{5} && \text{Simplify and group terms}\\ \\ &= 3 + i \end{aligned} \end{equation}$