Answer :
Sure, let's solve the problem step-by-step by using the distributive property, which involves multiplying each term in one polynomial by each term in the other.
Given expression to expand:
[tex]\[
(-2x - 9y^2)(-4x - 3)
\][/tex]
### Step 1: Distribute [tex]\(-2x\)[/tex]
1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
### Step 2: Distribute [tex]\(-9y^2\)[/tex]
1. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
2. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
### Step 3: Combine all the terms
Now, we combine all the terms we computed:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So the expanded expression, or the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
This matches the option:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Given expression to expand:
[tex]\[
(-2x - 9y^2)(-4x - 3)
\][/tex]
### Step 1: Distribute [tex]\(-2x\)[/tex]
1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
### Step 2: Distribute [tex]\(-9y^2\)[/tex]
1. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
2. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
### Step 3: Combine all the terms
Now, we combine all the terms we computed:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So the expanded expression, or the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
This matches the option:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
