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Multiply the expressions:

\[ (x^2 - 5x)(2x^2 + x - 3) \]

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

B. \[4x^4 + 9x^3 - 8x^2 + 15x\]

C. \[2x^4 - 9x^3 - 9x^2 - 15x\]

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

Answer :

To solve this problem, we need to multiply two polynomials: [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex]. Here is a detailed step-by-step solution:

1. Distribute [tex]\(x^2\)[/tex] Across [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, multiplying [tex]\(x^2\)[/tex] gives us [tex]\(2x^4 + x^3 - 3x^2\)[/tex].

2. Distribute [tex]\(-5x\)[/tex] Across [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, multiplying [tex]\(-5x\)[/tex] gives us [tex]\(-10x^3 - 5x^2 + 15x\)[/tex].

3. Combine Like Terms:

[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/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]

Now, putting it all together, the combined polynomial:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]

So, the correct result of multiplying these polynomials is [tex]\(\boxed{2x^4 - 9x^3 - 8x^2 + 15x}\)[/tex]. Hence, the answer to the problem is option D.

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