High School

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What is the product of the following 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 :

Sure! Let's find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] step by step.

### Step 1: Use the Distributive Property

The distributive property states that [tex]\(a(b + c) = ab + ac\)[/tex]. We will apply this property to distribute each term in the first parentheses to each term in the second parentheses.

1. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:

- [tex]\(-2x \times -4x = 8x^2\)[/tex]
- [tex]\(-2x \times -3 = 6x\)[/tex]

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

- [tex]\(-9y^2 \times -4x = 36xy^2\)[/tex]
- [tex]\(-9y^2 \times -3 = 27y^2\)[/tex]

### Step 2: Combine All Terms

Now, combine all the results:

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

Putting them together, the product of the expression is:

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

So the correct answer is:

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

This matches the option indicated as [tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex].

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