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Answer :
- Expand the product of the two binomials using the distributive property.
- Multiply each term: $(-2x)(-4x) = 8x^2$, $(-2x)(-3) = 6x$, $(-9y^2)(-4x) = 36xy^2$, $(-9y^2)(-3) = 27y^2$.
- Combine the terms: $8x^2 + 6x + 36xy^2 + 27y^2$.
- The product is $\boxed{8 x^2+6 x+36 x y^2+27 y^2}$.
### Explanation
1. Understanding the Problem
We are given the expression $\left(-2 x-9 y^2\right)(-4 x-3)$ and asked to find the product. We will use the distributive property (also known as FOIL) to expand the product of the two binomials.
2. Expanding the Product
Expanding the product:
$$\begin{aligned} \left(-2 x-9 y^2\right)(-4 x-3) &= (-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3) \\ &= 8x^2 + 6x + 36xy^2 + 27y^2 \end{aligned}$$
3. Identifying the Correct Option
Comparing the expanded expression $8x^2 + 6x + 36xy^2 + 27y^2$ with the given options, we find that it matches the third option.
4. Final Answer
Therefore, the product of $\left(-2 x-9 y^2\right)(-4 x-3)$ is $8 x^2+6 x+36 x y^2+27 y^2$.
### Examples
Understanding how to multiply binomials is essential in various fields, such as physics and engineering, where complex equations often need simplification. For instance, when calculating the area of a rectangular garden with sides expressed as binomials, multiplying these expressions gives the total area. This skill also helps in modeling growth rates or predicting outcomes in different scenarios, making it a fundamental tool in problem-solving.
- Multiply each term: $(-2x)(-4x) = 8x^2$, $(-2x)(-3) = 6x$, $(-9y^2)(-4x) = 36xy^2$, $(-9y^2)(-3) = 27y^2$.
- Combine the terms: $8x^2 + 6x + 36xy^2 + 27y^2$.
- The product is $\boxed{8 x^2+6 x+36 x y^2+27 y^2}$.
### Explanation
1. Understanding the Problem
We are given the expression $\left(-2 x-9 y^2\right)(-4 x-3)$ and asked to find the product. We will use the distributive property (also known as FOIL) to expand the product of the two binomials.
2. Expanding the Product
Expanding the product:
$$\begin{aligned} \left(-2 x-9 y^2\right)(-4 x-3) &= (-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3) \\ &= 8x^2 + 6x + 36xy^2 + 27y^2 \end{aligned}$$
3. Identifying the Correct Option
Comparing the expanded expression $8x^2 + 6x + 36xy^2 + 27y^2$ with the given options, we find that it matches the third option.
4. Final Answer
Therefore, the product of $\left(-2 x-9 y^2\right)(-4 x-3)$ is $8 x^2+6 x+36 x y^2+27 y^2$.
### Examples
Understanding how to multiply binomials is essential in various fields, such as physics and engineering, where complex equations often need simplification. For instance, when calculating the area of a rectangular garden with sides expressed as binomials, multiplying these expressions gives the total area. This skill also helps in modeling growth rates or predicting outcomes in different scenarios, making it a fundamental tool in problem-solving.
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