Answer :
Certainly! Let's work through the problem step-by-step to find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex].
### Step 1: Distribute Each Term
We will use the distributive property to expand the expression. This involves multiplying each term in the first parenthesis by each term in the second parenthesis.
#### Distribute [tex]\(-2x\)[/tex]:
1. [tex]\((-2x) \times (-4x) = 8x^2\)[/tex]
2. [tex]\((-2x) \times (-3) = 6x\)[/tex]
#### Distribute [tex]\(-9y^2\)[/tex]:
3. [tex]\((-9y^2) \times (-4x) = 36xy^2\)[/tex]
4. [tex]\((-9y^2) \times (-3) = 27y^2\)[/tex]
### Step 2: Combine All the Results
Now, we'll combine all these products to get the expanded expression:
- [tex]\(8x^2\)[/tex]
- [tex]\(6x\)[/tex]
- [tex]\(36xy^2\)[/tex]
- [tex]\(27y^2\)[/tex]
Putting it all together, the expanded expression is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
This matches one of the given options:
[tex]\[8 x^2 + 6 x + 36 x y^2 + 27 y^2\][/tex]
So, the correct answer is:
[tex]\[8 x^2 + 6 x + 36 x y^2 + 27 y^2\][/tex]
### Step 1: Distribute Each Term
We will use the distributive property to expand the expression. This involves multiplying each term in the first parenthesis by each term in the second parenthesis.
#### Distribute [tex]\(-2x\)[/tex]:
1. [tex]\((-2x) \times (-4x) = 8x^2\)[/tex]
2. [tex]\((-2x) \times (-3) = 6x\)[/tex]
#### Distribute [tex]\(-9y^2\)[/tex]:
3. [tex]\((-9y^2) \times (-4x) = 36xy^2\)[/tex]
4. [tex]\((-9y^2) \times (-3) = 27y^2\)[/tex]
### Step 2: Combine All the Results
Now, we'll combine all these products to get the expanded expression:
- [tex]\(8x^2\)[/tex]
- [tex]\(6x\)[/tex]
- [tex]\(36xy^2\)[/tex]
- [tex]\(27y^2\)[/tex]
Putting it all together, the expanded expression is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
This matches one of the given options:
[tex]\[8 x^2 + 6 x + 36 x y^2 + 27 y^2\][/tex]
So, the correct answer is:
[tex]\[8 x^2 + 6 x + 36 x y^2 + 27 y^2\][/tex]
