High School

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Multiply:

[tex]\left(x^2-5x\right)\left(2x^2+x-3\right)[/tex]

A. [tex]2x^4-9x^3-8x^2+15x[/tex]

B. [tex]2x^4+9x^3-8x^2+15x[/tex]

C. [tex]4x^4+9x^3-8x^2+15x[/tex]

D. [tex]2x^4-9x^3-9x^2-15x[/tex]

Answer :

To solve the problem of multiplying the two polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex], we follow these steps:

1. Distribute each term in the first polynomial to each term in the second polynomial.

- 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]

- 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]

2. Combine all the results from the multiplication:
- The products 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].

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]

4. Write the final expression:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]

By following these steps carefully, we find that the correct answer is:

A. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]

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