To factor the polynomial [tex]\(x^3 - 4x^2 + 7x - 28\)[/tex] by grouping, follow these steps:
1. Group the terms: Separate the polynomial into two groups. In this case, we separate it as:
[tex]\[
(x^3 - 4x^2) + (7x - 28)
\][/tex]
2. Factor out the Greatest Common Factor (GCF) from each group:
- For the first group [tex]\((x^3 - 4x^2)\)[/tex], the GCF is [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x - 4)
\][/tex]
- For the second group [tex]\((7x - 28)\)[/tex], the GCF is 7:
[tex]\[
7(x - 4)
\][/tex]
3. Factor by grouping: Since both groups contain the common binomial factor [tex]\((x - 4)\)[/tex], factor this binomial out:
[tex]\[
x^2(x - 4) + 7(x - 4) = (x - 4)(x^2 + 7)
\][/tex]
So, the polynomial [tex]\( x^3 - 4x^2 + 7x - 28 \)[/tex] factors to:
[tex]\[
(x^2 + 7)(x - 4)
\][/tex]
Therefore, the correct resulting expression is:
[tex]\[
\boxed{(x^2 + 7)(x - 4)}
\][/tex]
