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What is the product of the following expression?

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

A. \(-8x^2 - 6x - 36xy^2 - 27y^2\)

B. \(-14x^2 - 36xy^2 + 27y^2\)

C. \(8x^2 + 6x + 36xy^2 + 27y^2\)

D. \(14x^2 + 36xy^2 + 27y^2\)

Answer :

We begin with the product
$$
(-2x - 9y^2)(-4x - 3).
$$

To find the result, we use the distributive property (or FOIL method) to multiply each term in the first parenthesis by each term in the second.

1. Multiply the first terms:
$$
(-2x) \cdot (-4x) = 8x^2.
$$

2. Multiply the outer terms:
$$
(-2x) \cdot (-3) = 6x.
$$

3. Multiply the inner terms:
$$
(-9y^2) \cdot (-4x) = 36xy^2.
$$

4. Multiply the last terms:
$$
(-9y^2) \cdot (-3) = 27y^2.
$$

Now, add all these results together:
$$
8x^2 + 6x + 36xy^2 + 27y^2.
$$

Thus, the final product is
$$
\boxed{8x^2 + 6x + 36xy^2 + 27y^2}.
$$

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