College

What is the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex]?

A. [tex]\(-8x^2 - 6x - 36xy^2 - 27y^2\)[/tex]

B. [tex]\(-14x^2 - 36xy^2 + 27y^2\)[/tex]

C. [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex]

D. [tex]\(14x^2 + 36xy^2 + 27y^2\)[/tex]

Answer :

To find the product of the expressions [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex], we need to distribute each term in the first expression by each term in the second expression and then simplify the result.

Let's break it down step by step:

1. Distribute [tex]\(-2x\)[/tex] to both terms in [tex]\((-4x - 3)\)[/tex]:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]: [tex]\((-2x) \times (-4x) = 8x^2\)[/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]: [tex]\((-2x) \times (-3) = 6x\)[/tex]

2. Distribute [tex]\(-9y^2\)[/tex] to both terms in [tex]\((-4x - 3)\)[/tex]:
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]: [tex]\((-9y^2) \times (-4x) = 36xy^2\)[/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]: [tex]\((-9y^2) \times (-3) = 27y^2\)[/tex]

3. Combine all the terms we found:
- [tex]\(8x^2\)[/tex] from step 1
- [tex]\(6x\)[/tex] from step 1
- [tex]\(36xy^2\)[/tex] from step 2
- [tex]\(27y^2\)[/tex] from step 2

4. Write down the final expression:
- The expression is [tex]\(8x^2 + 36xy^2 + 6x + 27y^2\)[/tex]

This is the simplified product of [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex].

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