Answer :
To solve the problem of multiplying the two expressions [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex], we'll follow the distributive property step-by-step, also known as the "FOIL" method for binomials, but it's extended here to handle trinomials.
1. Distribute each term of the first expression:
- Multiply [tex]\(x^2\)[/tex] with each term in the second expression [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. Distribute the second term of the first expression [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] with each term in the second expression [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]
3. Combine all the terms obtained:
- Start by writing out all the terms from each distribution:
[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]
- The [tex]\(x^4\)[/tex] term and the [tex]\(x\)[/tex] term don't have like terms, so they remain the same.
5. Final result:
- Combine all terms to get: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
Thus, the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex] simplifies to [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
The correct choice from the options given is:
A. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
1. Distribute each term of the first expression:
- Multiply [tex]\(x^2\)[/tex] with each term in the second expression [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. Distribute the second term of the first expression [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] with each term in the second expression [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]
3. Combine all the terms obtained:
- Start by writing out all the terms from each distribution:
[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]
- The [tex]\(x^4\)[/tex] term and the [tex]\(x\)[/tex] term don't have like terms, so they remain the same.
5. Final result:
- Combine all terms to get: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
Thus, the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex] simplifies to [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
The correct choice from the options given is:
A. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
