Reduce `((1)/(1+2i)+(3)/(1-i))((3-2i)/(1+3i))` to the form `(a + ib).`

Explanation

To reduce the expression \(\left(\frac{1}{1+2i} + \frac{3}{1-i}\right)\left(\frac{3-2i}{1+3i}\right)\) to the form \(a + ib\), we will follow these steps:

### Step 1: Simplify \(\frac{1}{1+2i}\)

To simplify \(\frac{1}{1+2i}\), we multiply the numerator and the denominator by the conjugate of the denominator:

\[
\frac{1}{1+2i} \cdot \frac{1-2i}{1-2i} = \frac{1-2i}{(1+2i)(1-2i)} = \frac{1-2i}{1^2 - (2i)^2} = \frac{1-2i}{1 - (-4)} = \frac{1-2i}{5}
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