To solve the problem of finding the product [tex]\((x^2 + 3x + 1)(x^2 + x + 2)\)[/tex] and express the result in descending powers of [tex]\(x\)[/tex], follow these steps:
1. Multiply each term of the first polynomial by each term of the second polynomial:
- Start with [tex]\(x^2\)[/tex]:
[tex]\[
x^2 \cdot x^2 = x^4
\][/tex]
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
x^2 \cdot 2 = 2x^2
\][/tex]
- Next, with [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot x^2 = 3x^3
\][/tex]
[tex]\[
3x \cdot x = 3x^2
\][/tex]
[tex]\[
3x \cdot 2 = 6x
\][/tex]
- Finally, with [tex]\(1\)[/tex]:
[tex]\[
1 \cdot x^2 = x^2
\][/tex]
[tex]\[
1 \cdot x = x
\][/tex]
[tex]\[
1 \cdot 2 = 2
\][/tex]
2. Combine like terms:
- Collect all [tex]\(x^4\)[/tex] terms:
[tex]\[
x^4
\][/tex]
- Collect all [tex]\(x^3\)[/tex] terms:
[tex]\[
x^3 + 3x^3 = 4x^3
\][/tex]
- Collect all [tex]\(x^2\)[/tex] terms:
[tex]\[
2x^2 + 3x^2 + x^2 = 6x^2
\][/tex]
- Collect all [tex]\(x\)[/tex] terms:
[tex]\[
6x + x = 7x
\][/tex]
- Constant terms:
[tex]\[
2
\][/tex]
3. Write the final expanded expression in descending order:
[tex]\[
x^4 + 4x^3 + 6x^2 + 7x + 2
\][/tex]
Therefore, the product [tex]\((x^2 + 3x + 1)(x^2 + x + 2)\)[/tex] is:
[tex]\[
\boxed{x^4 + 4x^3 + 6x^2 + 7x + 2}
\][/tex]
