College

Multiply:

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

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

B. [tex]\[ 2x^4 - 9x^3 - 9x^2 - 15x \][/tex]

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

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

Answer :

Sure! Let's multiply the polynomials step-by-step to get the solution.

We need to multiply:
[tex]\[
(x^2 - 5x)(2x^2 + x - 3)
\][/tex]

### Step-by-Step Expansion:

1. Distribute [tex]\(x^2\)[/tex] to 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]

So, distributing [tex]\(x^2\)[/tex] gives:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]

2. Distribute [tex]\(-5x\)[/tex] to 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]

So, distributing [tex]\(-5x\)[/tex] gives:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]

3. Combine all the terms from the two distributions:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]

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

So, putting everything together, we have:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]

Hence, the correct answer is:
[tex]\[
\boxed{2x^4 - 9x^3 - 8x^2 + 15x}
\][/tex]

So the correct option is:
D. [tex]\(2 x^4 - 9 x^3 - 8 x^2 + 15 x\)[/tex].

Other Questions