College

What is the product of the expression?

[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 multiply the polynomials, we use the distributive property (also known as the FOIL method when there are two binomials). We have:

[tex]$$
(-2x - 9y^2)(-4x - 3).
$$[/tex]

Step 1. Multiply the first term of the first polynomial by each term of the second polynomial:

- Multiply [tex]$-2x$[/tex] by [tex]$-4x$[/tex]:
[tex]$$
(-2x) \cdot (-4x) = 8x^2.
$$[/tex]
- Multiply [tex]$-2x$[/tex] by [tex]$-3$[/tex]:
[tex]$$
(-2x) \cdot (-3) = 6x.
$$[/tex]

Step 2. Multiply the second term of the first polynomial by each term of the second polynomial:

- Multiply [tex]$-9y^2$[/tex] by [tex]$-4x$[/tex]:
[tex]$$
(-9y^2) \cdot (-4x) = 36xy^2.
$$[/tex]
- Multiply [tex]$-9y^2$[/tex] by [tex]$-3$[/tex]:
[tex]$$
(-9y^2) \cdot (-3) = 27y^2.
$$[/tex]

Step 3. Combine all the terms to get the final product:

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

Thus, the product is

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