College

What is the product?

[tex]\left(-2x - 9y^2\right)(-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 expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will apply the distributive property. Here's the detailed, step-by-step solution:

1. Apply the Distributive Property:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
In our expression, the terms to distribute are [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex].

2. Distribute each term in the first polynomial to each term in the second polynomial:

- 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]

- 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:
- After distributing, add all the products together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

Therefore, the expanded product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

This matches with the third option provided in the choices:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]