Answer :
Sure! Let's solve the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex] step by step to match it with one of the given choices.
Step 1: Distribute each term from the first polynomial to each term in the second polynomial.
We'll do this by multiplying each term in the first polynomial, [tex]\((x^2 - 5x)\)[/tex], with each term in the second polynomial, [tex]\((2x^2 + x - 3)\)[/tex].
1. Multiply [tex]\(x^2\)[/tex] by 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]
2. Multiply [tex]\(-5x\)[/tex] by 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]
Step 2: Combine all the terms from these multiplications.
- The terms we obtained are:
[tex]\(2x^4\)[/tex],
[tex]\(x^3\)[/tex],
[tex]\(-3x^2\)[/tex],
[tex]\(-10x^3\)[/tex],
[tex]\(-5x^2\)[/tex],
and [tex]\(15x\)[/tex].
Step 3: 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]
- Now, put all combined terms together:
[tex]\[
2x^4 + (-9x^3) + (-8x^2) + 15x = 2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Conclusion:
The expanded expression is [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]. This matches with choice B:
B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
So, the correct answer is B.
Step 1: Distribute each term from the first polynomial to each term in the second polynomial.
We'll do this by multiplying each term in the first polynomial, [tex]\((x^2 - 5x)\)[/tex], with each term in the second polynomial, [tex]\((2x^2 + x - 3)\)[/tex].
1. Multiply [tex]\(x^2\)[/tex] by 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]
2. Multiply [tex]\(-5x\)[/tex] by 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]
Step 2: Combine all the terms from these multiplications.
- The terms we obtained are:
[tex]\(2x^4\)[/tex],
[tex]\(x^3\)[/tex],
[tex]\(-3x^2\)[/tex],
[tex]\(-10x^3\)[/tex],
[tex]\(-5x^2\)[/tex],
and [tex]\(15x\)[/tex].
Step 3: 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]
- Now, put all combined terms together:
[tex]\[
2x^4 + (-9x^3) + (-8x^2) + 15x = 2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
Conclusion:
The expanded expression is [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]. This matches with choice B:
B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
So, the correct answer is B.
