High School

What is the product?

\[ [tex] (-2x - 9y^2)(-4x - 3) \]

A. [tex] -14x^2 - 36xy^2 + 27y^2 [/tex]

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

C. [tex] 14x^2 + 36xy^2 + 27y^2 [/tex]

D. [tex] -8x^2 - 6x - 36xy^2 - 27y^2 [/tex]

Answer :

To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will use the distributive property. This property allows us to multiply each term in the first expression by each term in the second expression. Let's go through the process step-by-step:

1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:

[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]

Here, the negative signs cancel each other out, and we multiply the coefficients to get 8. The [tex]\(x\)[/tex] terms combine to [tex]\(x^2\)[/tex] because we are multiplying [tex]\(x \times x\)[/tex].

2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:

[tex]\[
(-2x) \times (-3) = 6x
\][/tex]

Again, the negative signs cancel, and we are left with a positive product. Multiply the coefficients [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex] to get 6, and the [tex]\(x\)[/tex] remains as it is.

3. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:

[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]

The negative signs cancel each other. Multiply the coefficients [tex]\(-9\)[/tex] and [tex]\(-4\)[/tex] to get 36. We have [tex]\(x\)[/tex] with [tex]\(y^2\)[/tex], which becomes [tex]\(xy^2\)[/tex].

4. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:

[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]

Again, the negative signs cancel, and multiply the coefficients [tex]\(-9\)[/tex] and [tex]\(-3\)[/tex] to get 27.

Now, combine all these results to write the complete polynomial:

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

So, the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:

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

This corresponds to the option:

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