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Answer :
To solve the problem [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex], we will multiply the two polynomials together by distributing each term in the first polynomial to every term in the second polynomial. Here is a detailed step-by-step solution:
1. Distribute [tex]\(x^2\)[/tex] in [tex]\(x^2 - 5x\)[/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]
Adding these together, we get:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] in [tex]\(x^2 - 5x\)[/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]
Adding these together, we get:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Combine all the terms from the distributions:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
4. Combine like terms:
- 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, the expression now is:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Now let's match this to the multiple choice options:
- [tex]\(A. \ 2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
- [tex]\(B. \ 2x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
- [tex]\(C. \ 4x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
- [tex]\(D. \ 2x^4 - 9x^3 - 9x^2 - 15x\)[/tex]
The correct answer is [tex]\( \boxed{A} \)[/tex].
1. Distribute [tex]\(x^2\)[/tex] in [tex]\(x^2 - 5x\)[/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]
Adding these together, we get:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] in [tex]\(x^2 - 5x\)[/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]
Adding these together, we get:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Combine all the terms from the distributions:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
4. Combine like terms:
- 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, the expression now is:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Now let's match this to the multiple choice options:
- [tex]\(A. \ 2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
- [tex]\(B. \ 2x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
- [tex]\(C. \ 4x^4 + 9x^3 - 8x^2 + 15x\)[/tex]
- [tex]\(D. \ 2x^4 - 9x^3 - 9x^2 - 15x\)[/tex]
The correct answer is [tex]\( \boxed{A} \)[/tex].
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