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 :

Sure! Let's solve the problem step by step.

We are asked to find the product of [tex]\(\left(-2x - 9y^2\right)(-4x - 3)\)[/tex].

1. Distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[
(-2x - 9y^2)(-4x - 3) = (-2x) \cdot (-4x) + (-2x) \cdot (-3) + (-9y^2) \cdot (-4x) + (-9y^2) \cdot (-3)
\][/tex]

2. Multiply each pair of terms:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]

3. Combine all these terms:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

So, the product [tex]\(\left(-2x - 9y^2\right)(-4x - 3)\)[/tex] simplifies to:

[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]

From the given choices, the correct answer is:
[tex]\[ \boxed{8x^2 + 6x + 36x y^2 + 27y^2} \][/tex]