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Answer :
Let's solve the problem of multiplying the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex] step by step.
1. Distribute [tex]\(x^2\)[/tex] across the second polynomial:
[tex]\[
x^2 \cdot (2x^2 + x - 3) = x^2 \cdot 2x^2 + x^2 \cdot x + x^2 \cdot (-3)
\][/tex]
This simplifies to:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] across the second polynomial:
[tex]\[
-5x \cdot (2x^2 + x - 3) = -5x \cdot 2x^2 + -5x \cdot x + -5x \cdot (-3)
\][/tex]
This simplifies to:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Add the results together:
Combine like terms from the two distributions:
[tex]\[
(2x^4 + x^3 - 3x^2) + (-10x^3 - 5x^2 + 15x)
\][/tex]
Combine like terms:
- [tex]\(2x^4\)[/tex] (no like terms)
- [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- [tex]\((-3x^2) + (-5x^2) = -8x^2\)[/tex]
- [tex]\(15x\)[/tex] (no like terms)
4. Final expression:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
So the correct answer is B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
1. Distribute [tex]\(x^2\)[/tex] across the second polynomial:
[tex]\[
x^2 \cdot (2x^2 + x - 3) = x^2 \cdot 2x^2 + x^2 \cdot x + x^2 \cdot (-3)
\][/tex]
This simplifies to:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] across the second polynomial:
[tex]\[
-5x \cdot (2x^2 + x - 3) = -5x \cdot 2x^2 + -5x \cdot x + -5x \cdot (-3)
\][/tex]
This simplifies to:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Add the results together:
Combine like terms from the two distributions:
[tex]\[
(2x^4 + x^3 - 3x^2) + (-10x^3 - 5x^2 + 15x)
\][/tex]
Combine like terms:
- [tex]\(2x^4\)[/tex] (no like terms)
- [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- [tex]\((-3x^2) + (-5x^2) = -8x^2\)[/tex]
- [tex]\(15x\)[/tex] (no like terms)
4. Final expression:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
So the correct answer is B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
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