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Answer :
- Expand the product of the two binomials using the distributive property (FOIL method).
- Simplify each term after expansion.
- Combine the simplified terms to obtain the expanded form.
- 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: $(-2x - 9y^2)$ and $(-4x - 3)$. We will use the distributive property (also known as FOIL) to expand the product.
2. Expanding the Product
We need to multiply each term in the first binomial by each term in the second binomial:
$(-2x - 9y^2)(-4x - 3) = (-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3)$
3. Simplifying Each Term
Now, let's simplify each term:
* $(-2x)(-4x) = 8x^2$
* $(-2x)(-3) = 6x$
* $(-9y^2)(-4x) = 36xy^2$
* $(-9y^2)(-3) = 27y^2$
4. Combining Terms
Combine the simplified terms to get the expanded form of the product:
$8x^2 + 6x + 36xy^2 + 27y^2$
5. Identifying the Correct Answer
Comparing our result with the given options, we find that the correct answer is:
$8x^2 + 6x + 36xy^2 + 27y^2$
6. Final Answer
Therefore, the product of the given binomials is $8x^2 + 6x + 36xy^2 + 27y^2$.
### Examples
Understanding how to expand binomial products 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, expanding the product helps determine the total area in terms of its components. This skill is also crucial in economics for modeling revenue and cost functions, providing a clear understanding of how different factors interact and contribute to the overall outcome. Mastering binomial expansion enhances problem-solving abilities in real-world scenarios.
- Simplify each term after expansion.
- Combine the simplified terms to obtain the expanded form.
- 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: $(-2x - 9y^2)$ and $(-4x - 3)$. We will use the distributive property (also known as FOIL) to expand the product.
2. Expanding the Product
We need to multiply each term in the first binomial by each term in the second binomial:
$(-2x - 9y^2)(-4x - 3) = (-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3)$
3. Simplifying Each Term
Now, let's simplify each term:
* $(-2x)(-4x) = 8x^2$
* $(-2x)(-3) = 6x$
* $(-9y^2)(-4x) = 36xy^2$
* $(-9y^2)(-3) = 27y^2$
4. Combining Terms
Combine the simplified terms to get the expanded form of the product:
$8x^2 + 6x + 36xy^2 + 27y^2$
5. Identifying the Correct Answer
Comparing our result with the given options, we find that the correct answer is:
$8x^2 + 6x + 36xy^2 + 27y^2$
6. Final Answer
Therefore, the product of the given binomials is $8x^2 + 6x + 36xy^2 + 27y^2$.
### Examples
Understanding how to expand binomial products 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, expanding the product helps determine the total area in terms of its components. This skill is also crucial in economics for modeling revenue and cost functions, providing a clear understanding of how different factors interact and contribute to the overall outcome. Mastering binomial expansion enhances problem-solving abilities in real-world scenarios.
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