To multiply the polynomials [tex]\((5x^2 + x - 4)\)[/tex] and [tex]\((x + 2)\)[/tex], follow these steps:
1. Distribute each term of [tex]\(5x^2 + x - 4\)[/tex] with each term of [tex]\(x + 2\)[/tex]:
- First, distribute [tex]\(5x^2\)[/tex] over [tex]\(x + 2\)[/tex]:
[tex]\[
5x^2 \cdot x + 5x^2 \cdot 2 = 5x^3 + 10x^2
\][/tex]
- Next, distribute [tex]\(x\)[/tex] over [tex]\(x + 2\)[/tex]:
[tex]\[
x \cdot x + x \cdot 2 = x^2 + 2x
\][/tex]
- Finally, distribute [tex]\(-4\)[/tex] over [tex]\(x + 2\)[/tex]:
[tex]\[
-4 \cdot x + (-4) \cdot 2 = -4x - 8
\][/tex]
2. Combine all the products:
[tex]\[
(5x^3 + 10x^2) + (x^2 + 2x) + (-4x - 8)
\][/tex]
3. Combine like terms (group and simplify):
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
10x^2 + x^2 = 11x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
2x - 4x = -2x
\][/tex]
4. Write the final expression:
[tex]\[
5x^3 + 11x^2 -2x - 8
\][/tex]
Therefore, the product of the polynomials [tex]\((5x^2 + x - 4)\)[/tex] and [tex]\((x + 2)\)[/tex] is:
[tex]\[
\boxed{5x^3 + 11x^2 - 2x - 8}
\][/tex]
Comparing it to the given answer choices, the correct answer is:
[tex]\[
5x^3 + 11x^2 - 2x - 8
\][/tex]
