Answer :
- Multiply the first terms: $(-2x) \times (-4x) = 8x^2$.
- Multiply the outer terms: $(-2x) \times (-3) = 6x$.
- Multiply the inner terms: $(-9y^2) \times (-4x) = 36xy^2$.
- Multiply the last terms: $(-9y^2) \times (-3) = 27y^2$. 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: $(-2x - 9y^2)(-4x - 3)$. We will use the distributive property (also known as the FOIL method) to expand this product.
2. Multiplying the first terms
First, we multiply the first terms in each binomial: $(-2x) \times (-4x) = 8x^2$.
3. Multiplying the outer terms
Next, we multiply the outer terms: $(-2x) \times (-3) = 6x$.
4. Multiplying the inner terms
Then, we multiply the inner terms: $(-9y^2) \times (-4x) = 36xy^2$.
5. Multiplying the last terms
Finally, we multiply the last terms: $(-9y^2) \times (-3) = 27y^2$.
6. Combining the terms
Now, we combine all the terms we found: $8x^2 + 6x + 36xy^2 + 27y^2$.
7. Final Answer
Comparing this result with the given options, we see that it matches the third option: $8x^2 + 6x + 36xy^2 + 27y^2$. Therefore, the correct product is $8x^2 + 6x + 36xy^2 + 27y^2$.
### 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 helps determine the total area. This skill is also crucial in economics for modeling cost and revenue functions, where binomial multiplication can represent the interaction of different economic factors. Mastering binomial multiplication enhances problem-solving abilities in real-world scenarios.
- Multiply the outer terms: $(-2x) \times (-3) = 6x$.
- Multiply the inner terms: $(-9y^2) \times (-4x) = 36xy^2$.
- Multiply the last terms: $(-9y^2) \times (-3) = 27y^2$. 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: $(-2x - 9y^2)(-4x - 3)$. We will use the distributive property (also known as the FOIL method) to expand this product.
2. Multiplying the first terms
First, we multiply the first terms in each binomial: $(-2x) \times (-4x) = 8x^2$.
3. Multiplying the outer terms
Next, we multiply the outer terms: $(-2x) \times (-3) = 6x$.
4. Multiplying the inner terms
Then, we multiply the inner terms: $(-9y^2) \times (-4x) = 36xy^2$.
5. Multiplying the last terms
Finally, we multiply the last terms: $(-9y^2) \times (-3) = 27y^2$.
6. Combining the terms
Now, we combine all the terms we found: $8x^2 + 6x + 36xy^2 + 27y^2$.
7. Final Answer
Comparing this result with the given options, we see that it matches the third option: $8x^2 + 6x + 36xy^2 + 27y^2$. Therefore, the correct product is $8x^2 + 6x + 36xy^2 + 27y^2$.
### 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 helps determine the total area. This skill is also crucial in economics for modeling cost and revenue functions, where binomial multiplication can represent the interaction of different economic factors. Mastering binomial multiplication enhances problem-solving abilities in real-world scenarios.
