Let's find the expression for [tex]\(A \cdot B + C\)[/tex] given the polynomials [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
1. Define the polynomials:
- [tex]\(A = x + 1\)[/tex]
- [tex]\(B = x^2 + 2x - 1\)[/tex]
- [tex]\(C = 2x\)[/tex]
2. Calculate [tex]\(A \cdot B\)[/tex]:
[tex]\[
A \cdot B = (x + 1)(x^2 + 2x - 1)
\][/tex]
Use the distributive property to expand this product:
[tex]\[
(x + 1)(x^2 + 2x - 1) = x \cdot (x^2 + 2x - 1) + 1 \cdot (x^2 + 2x - 1)
\][/tex]
Compute each term:
[tex]\[
x \cdot (x^2 + 2x - 1) = x^3 + 2x^2 - x
\][/tex]
[tex]\[
1 \cdot (x^2 + 2x - 1) = x^2 + 2x - 1
\][/tex]
Combine these results:
[tex]\[
x^3 + 2x^2 - x + x^2 + 2x - 1 = x^3 + 3x^2 + x - 1
\][/tex]
3. Add [tex]\(C\)[/tex] to the result:
[tex]\[
A \cdot B + C = (x^3 + 3x^2 + x - 1) + 2x
\][/tex]
Combine like terms:
[tex]\[
x^3 + 3x^2 + x + 2x - 1 = x^3 + 3x^2 + 3x - 1
\][/tex]
Thus, the simplest form of [tex]\(A \cdot B + C\)[/tex] is:
[tex]\[
\boxed{x^3 + 3x^2 + 3x - 1}
\][/tex]
This matches one of the provided choices:
- [tex]\(x^3 + 3 x - 1\)[/tex]
- [tex]\(x^3 + 4 x - 1\)[/tex]
- [tex]\(x^3 + 3 x^2 + 3 x - 1\)[/tex]
- [tex]\(x^3 + 2 x^2 - x + 1\)[/tex]
So, the correct answer is:
[tex]\[
x^3 + 3 x^2 + 3 x - 1
\][/tex]
