To find the product of the two complex numbers [tex]\((1+i)(2+i)\)[/tex], follow these steps:
1. Distribute the terms using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(1+i)(2+i) = 1 \cdot 2 + 1 \cdot i + i \cdot 2 + i \cdot i
\][/tex]
2. Calculate each term of the product:
- [tex]\(1 \cdot 2 = 2\)[/tex]
- [tex]\(1 \cdot i = i\)[/tex]
- [tex]\(i \cdot 2 = 2i\)[/tex]
- [tex]\(i \cdot i = i^2\)[/tex]
3. Combine these results:
[tex]\[
2 + i + 2i + i^2
\][/tex]
4. Simplify the expression:
- Combine the like terms involving [tex]\(i\)[/tex]: [tex]\(i + 2i = 3i\)[/tex]
- Recall that [tex]\(i^2 = -1\)[/tex].
So the expression becomes:
[tex]\[
2 + 3i + (-1)
\][/tex]
5. Simplify further by combining the real parts:
[tex]\[
2 - 1 + 3i = 1 + 3i
\][/tex]
Therefore, the product [tex]\((1+i)(2+i)\)[/tex] is:
[tex]\[
\boxed{1 + 3i}
\][/tex]
The correct answer is:
E. [tex]\(\quad 1+3 i\)[/tex]
