Let's factor the polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] by grouping. Here are the steps involved in the factorization process:
1. Group the terms in pairs:
[tex]\[
x^3 + x^2 + x + 1 = (x^3 + x^2) + (x + 1)
\][/tex]
2. Factor out the common factors in each group:
[tex]\[
= x^2(x + 1) + 1(x + 1)
\][/tex]
3. Notice that [tex]\(x + 1\)[/tex] is a common factor in both groups:
[tex]\[
= (x^2 + 1)(x + 1)
\][/tex]
The polynomial [tex]\(x^3 + x^2 + x + 1\)[/tex] factors into [tex]\((x^2 + 1)(x + 1)\)[/tex].
Thus, the resulting expression is:
[tex]\[
\boxed{(x^2 + 1)(x + 1)}
\][/tex]
So, from the given options, the correct answer is:
[tex]\[
\left(x^2+1\right)(x+1)
\][/tex]
