Answer :
Sure! Let's find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] step by step.
### Step 1: Use the Distributive Property
The distributive property states that [tex]\(a(b + c) = ab + ac\)[/tex]. We will apply this property to distribute each term in the first parentheses to each term in the second parentheses.
1. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\(-2x \times -4x = 8x^2\)[/tex]
- [tex]\(-2x \times -3 = 6x\)[/tex]
2. Multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\(-9y^2 \times -4x = 36xy^2\)[/tex]
- [tex]\(-9y^2 \times -3 = 27y^2\)[/tex]
### Step 2: Combine All Terms
Now, combine all the results:
- [tex]\(8x^2\)[/tex]
- [tex]\(6x\)[/tex]
- [tex]\(36xy^2\)[/tex]
- [tex]\(27y^2\)[/tex]
Putting them together, the product of the expression is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
So the correct answer is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
This matches the option indicated as [tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex].
### Step 1: Use the Distributive Property
The distributive property states that [tex]\(a(b + c) = ab + ac\)[/tex]. We will apply this property to distribute each term in the first parentheses to each term in the second parentheses.
1. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\(-2x \times -4x = 8x^2\)[/tex]
- [tex]\(-2x \times -3 = 6x\)[/tex]
2. Multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\(-9y^2 \times -4x = 36xy^2\)[/tex]
- [tex]\(-9y^2 \times -3 = 27y^2\)[/tex]
### Step 2: Combine All Terms
Now, combine all the results:
- [tex]\(8x^2\)[/tex]
- [tex]\(6x\)[/tex]
- [tex]\(36xy^2\)[/tex]
- [tex]\(27y^2\)[/tex]
Putting them together, the product of the expression is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
So the correct answer is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
This matches the option indicated as [tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex].
