College

Multiply the following expressions:

[tex]
\[
\begin{array}{r}
x^2+4x+2 \\
\times \quad 2x^2+3x-4 \\
\hline
\end{array}
\]
[/tex]

Choose the correct option:

A. [tex]2x^4 + 12x^2 - 8[/tex]

B. [tex]2x^4 + 23x^2 - 10x - 8[/tex]

C. [tex]2x^4 + 11x^3 + 12x^2 - 10x - 8[/tex]

D. [tex]3x^2 + 7x - 2[/tex]

Answer :

To multiply the polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex], we need to distribute each term in the first polynomial by each term in the second polynomial and combine like terms afterward.

Here's a step-by-step explanation of the multiplication:

1. Multiply each term by [tex]\(x^2\)[/tex]:
- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \times 3x = 3x^3\)[/tex]
- [tex]\(x^2 \times (-4) = -4x^2\)[/tex]

2. Multiply each term by [tex]\(4x\)[/tex]:
- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(4x \times (-4) = -16x\)[/tex]

3. Multiply each term by [tex]\(2\)[/tex]:
- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]
- [tex]\(2 \times (-4) = -8\)[/tex]

4. Combine all the terms:
- Collect all terms from the products above:
[tex]\[2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8\][/tex]

5. Combine like terms:
- [tex]\(2x^4\)[/tex] (only one [tex]\(x^4\)[/tex])
- [tex]\((3x^3 + 8x^3) = 11x^3\)[/tex]
- [tex]\((-4x^2 + 12x^2 + 4x^2) = 12x^2\)[/tex]
- [tex]\((-16x + 6x) = -10x\)[/tex]
- [tex]\(-8\)[/tex] (only one constant term)

The final result, combining all like terms, is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]

This matches with option C. So, the correct answer is C. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].

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