Answer :
To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will use the distributive property, which involves multiplying each term in the first parenthesis by each term in the second parenthesis.
Let's break it down step-by-step:
1. Multiply the first terms:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
This is because multiplying two negative numbers gives a positive product.
2. Multiply the outer terms:
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]
Again, both numbers are negative, resulting in a positive product.
3. Multiply the inner terms:
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
The negative sign on both numbers gives a positive product.
4. Multiply the last terms:
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]
This gives a positive result for the same reason.
Now, combine all the products obtained:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Comparing this result with the options given in the question:
- The expression matches the option [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{8x^2 + 6x + 36xy^2 + 27y^2} \][/tex]
Let's break it down step-by-step:
1. Multiply the first terms:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
This is because multiplying two negative numbers gives a positive product.
2. Multiply the outer terms:
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]
Again, both numbers are negative, resulting in a positive product.
3. Multiply the inner terms:
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
The negative sign on both numbers gives a positive product.
4. Multiply the last terms:
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]
This gives a positive result for the same reason.
Now, combine all the products obtained:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Comparing this result with the options given in the question:
- The expression matches the option [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{8x^2 + 6x + 36xy^2 + 27y^2} \][/tex]
