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Answer :
- Multiply the first terms: $(-2x)(-4x) = 8x^2$.
- Multiply the outer terms: $(-2x)(-3) = 6x$.
- Multiply the inner terms: $(-9y^2)(-4x) = 36xy^2$.
- Multiply the last terms: $(-9y^2)(-3) = 27y^2$. The product is $\boxed{8 x^2+6 x+36 x y^2+27 y^2}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of two binomials: $(-2x - 9y^2)(-4x - 3)$. To do this, we will use the distributive property (also known as the FOIL method).
2. Applying the Distributive Property
We multiply each term in the first binomial by each term in the second binomial:
$(-2x)(-4x) = 8x^2$
$(-2x)(-3) = 6x$
$(-9y^2)(-4x) = 36xy^2$
$(-9y^2)(-3) = 27y^2$
3. Combining the Terms
Now, we add all the terms together:
$8x^2 + 6x + 36xy^2 + 27y^2$
4. Identifying the Correct Answer
Comparing our result with the given options, we find that the correct answer is:
$8x^2 + 6x + 36xy^2 + 27y^2$
### Examples
Understanding how to multiply binomials is useful in many areas, such as calculating areas of rectangles with variable side lengths, or in physics when dealing with polynomial expressions for motion or energy. For example, if you have a rectangular garden where the length is $(-2x - 9y^2)$ meters and the width is $(-4x - 3)$ meters, finding the area involves multiplying these two expressions together. The result, $8x^2 + 6x + 36xy^2 + 27y^2$ square meters, gives you the total area of the garden in terms of $x$ and $y$.
- Multiply the outer terms: $(-2x)(-3) = 6x$.
- Multiply the inner terms: $(-9y^2)(-4x) = 36xy^2$.
- Multiply the last terms: $(-9y^2)(-3) = 27y^2$. The product is $\boxed{8 x^2+6 x+36 x y^2+27 y^2}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of two binomials: $(-2x - 9y^2)(-4x - 3)$. To do this, we will use the distributive property (also known as the FOIL method).
2. Applying the Distributive Property
We multiply each term in the first binomial by each term in the second binomial:
$(-2x)(-4x) = 8x^2$
$(-2x)(-3) = 6x$
$(-9y^2)(-4x) = 36xy^2$
$(-9y^2)(-3) = 27y^2$
3. Combining the Terms
Now, we add all the terms together:
$8x^2 + 6x + 36xy^2 + 27y^2$
4. Identifying the Correct Answer
Comparing our result with the given options, we find that the correct answer is:
$8x^2 + 6x + 36xy^2 + 27y^2$
### Examples
Understanding how to multiply binomials is useful in many areas, such as calculating areas of rectangles with variable side lengths, or in physics when dealing with polynomial expressions for motion or energy. For example, if you have a rectangular garden where the length is $(-2x - 9y^2)$ meters and the width is $(-4x - 3)$ meters, finding the area involves multiplying these two expressions together. The result, $8x^2 + 6x + 36xy^2 + 27y^2$ square meters, gives you the total area of the garden in terms of $x$ and $y$.
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