To find the expression [tex]\( A B + C \)[/tex] in its simplest form, we have to follow a series of polynomial operations on the given polynomials [tex]\( A = x + 1 \)[/tex], [tex]\( B = x^2 + 2x - 1 \)[/tex], and [tex]\( C = 2x \)[/tex].
1. First, find the product [tex]\( A \cdot B \)[/tex].
[tex]\[
A = x + 1
\][/tex]
[tex]\[
B = x^2 + 2x - 1
\][/tex]
To get [tex]\( A \cdot B \)[/tex], distribute each term in [tex]\( A \)[/tex] through [tex]\( B \)[/tex]:
[tex]\[
A \cdot B = (x + 1)(x^2 + 2x - 1)
\][/tex]
This can be expanded as follows:
[tex]\[
= x(x^2 + 2x - 1) + 1(x^2 + 2x - 1)
\][/tex]
Simplifying each part:
[tex]\[
= x^3 + 2x^2 - x + x^2 + 2x - 1
\][/tex]
Combine like terms:
[tex]\[
= x^3 + 3x^2 + x - 1
\][/tex]
2. Next, add the polynomial [tex]\( C \)[/tex] to the product [tex]\( A \cdot B \)[/tex]:
[tex]\[
C = 2x
\][/tex]
So, add [tex]\( 2x \)[/tex] to [tex]\( x^3 + 3x^2 + x - 1 \)[/tex]:
[tex]\[
A \cdot B + C = x^3 + 3x^2 + x - 1 + 2x
\][/tex]
Combining the like terms:
[tex]\[
= x^3 + 3x^2 + 3x - 1
\][/tex]
Therefore, the simplified form of the expression [tex]\( A B + C \)[/tex] is:
[tex]\[
x^3 + 3x^2 + 3x - 1
\][/tex]
Among the given choices, the correct answer is:
[tex]\[
\boxed{x^3 + 3x^2 + 3x - 1}
\][/tex]
