College

Multiply:

[tex]\left(x^2-5x\right)\left(2x^2+x-3\right)[/tex]

A. [tex]4x^4+9x^3-8x^2+15x[/tex]
B. [tex]2x^4-9x^3-9x^2-15x[/tex]
C. [tex]2x^4+9x^3-8x^2+15x[/tex]
D. [tex]2x^4-9x^3-8x^2+15x[/tex]

Answer :

Sure! Let's multiply the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex] step by step to find the correct answer.

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

2. Distribute [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by each term in [tex]\(2x^2 + x - 3\)[/tex].
[tex]\[
-5x \cdot 2x^2 = -10x^3
\][/tex]
[tex]\[
-5x \cdot x = -5x^2
\][/tex]
[tex]\[
-5x \cdot (-3) = 15x
\][/tex]

3. Combine all the terms:
- Add all the results from steps 1 and 2 together.
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]

4. Combine like terms:
- Combine all the terms with the same power of [tex]\(x\)[/tex].
[tex]\[
2x^4 + (1x^3 - 10x^3) + (-3x^2 - 5x^2) + 15x
\][/tex]
[tex]\[
= 2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]

Therefore, the product of the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex] is [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].

The correct answer is D. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].

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