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------------------------------------------------ Multiply the expressions:

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

Choose the correct option:

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

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

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

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

To multiply the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex], we'll use the distributive property (also known as the FOIL method when dealing with binomials). Here is a step-by-step breakdown:

1. Distribute [tex]\(x^2\)[/tex] from the first polynomial to each term in 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]

[tex]\[
= 2x^4 + x^3 - 3x^2
\][/tex]

2. Distribute [tex]\(-5x\)[/tex] from the first polynomial to each term in the second polynomial:

[tex]\[
-5x \cdot (2x^2 + x - 3) = -5x \cdot 2x^2 - 5x \cdot x + 5x \cdot 3
\][/tex]

[tex]\[
= -10x^3 - 5x^2 + 15x
\][/tex]

3. Add the results of the two distributions together:

[tex]\[
(2x^4 + x^3 - 3x^2) + (-10x^3 - 5x^2 + 15x)
\][/tex]

4. Combine like terms:

- For [tex]\(x^4\)[/tex], we have: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex], we combine [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex], we combine [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- For [tex]\(x\)[/tex], we have: [tex]\(15x\)[/tex]

5. Write the final expression:

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

The correct answer is B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].