To determine which of the provided expressions is equivalent to [tex]\((x+7)\left(x^2-3x+2\right)\)[/tex], we should expand and simplify the given expression step by step.
Given expression:
[tex]\[
(x+7)(x^2 - 3x + 2)
\][/tex]
We will use the distributive property (also known as the FOIL method for binomials) to expand the expression:
1. Distribute [tex]\(x\)[/tex] from [tex]\( (x + 7) \)[/tex] to each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
[tex]\[
x \cdot x^2 + x \cdot (-3x) + x \cdot 2 = x^3 - 3x^2 + 2x
\][/tex]
2. Distribute [tex]\(7\)[/tex] from [tex]\( (x + 7) \)[/tex] to each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
[tex]\[
7 \cdot x^2 + 7 \cdot (-3x) + 7 \cdot 2 = 7x^2 - 21x + 14
\][/tex]
3. Combine the results from both distributions:
[tex]\[
(x^3 - 3x^2 + 2x) + (7x^2 - 21x + 14)
\][/tex]
4. Combine like terms:
[tex]\[
x^3 + (-3x^2 + 7x^2) + (2x - 21x) + 14
\][/tex]
[tex]\[
x^3 + 4x^2 - 19x + 14
\][/tex]
Through this step-by-step expansion and simplification, we determine that the expression [tex]\((x + 7)(x^2 - 3x + 2)\)[/tex] is equivalent to:
[tex]\[
x^3 + 4x^2 - 19x + 14
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B. \, x^3 + 4 x^2 - 19 x + 14}
\][/tex]
