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.
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.
