Answer :
Certainly! Let's solve the problem step-by-step by expanding the expression [tex]\(( -2x - 9y^2 ) ( -4x - 3 )\)[/tex].
Step 1: Apply the Distributive Property
We need to distribute each term in the first parenthesis across each term in the second parenthesis:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]
Step 2: Perform Each Multiplication
1. [tex]\((-2x) \cdot (-4x)\)[/tex]:
[tex]\(-2 \times -4 = 8\)[/tex], and [tex]\(x \times x = x^2\)[/tex]
Result: [tex]\(8x^2\)[/tex]
2. [tex]\((-2x) \cdot (-3)\)[/tex]:
[tex]\(-2 \times -3 = 6\)[/tex], and the variable is [tex]\(x\)[/tex]
Result: [tex]\(6x\)[/tex]
3. [tex]\((-9y^2) \cdot (-4x)\)[/tex]:
[tex]\(-9 \times -4 = 36\)[/tex], and the variables [tex]\(y^2 \times x\)[/tex]
Result: [tex]\(36xy^2\)[/tex]
4. [tex]\((-9y^2) \cdot (-3)\)[/tex]:
[tex]\(-9 \times -3 = 27\)[/tex], and the variable is [tex]\(y^2\)[/tex]
Result: [tex]\(27y^2\)[/tex]
Step 3: Combine All the Terms
Now, combine all the individual results:
[tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex]
This matches one of the provided options:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the answer is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
Step 1: Apply the Distributive Property
We need to distribute each term in the first parenthesis across each term in the second parenthesis:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]
Step 2: Perform Each Multiplication
1. [tex]\((-2x) \cdot (-4x)\)[/tex]:
[tex]\(-2 \times -4 = 8\)[/tex], and [tex]\(x \times x = x^2\)[/tex]
Result: [tex]\(8x^2\)[/tex]
2. [tex]\((-2x) \cdot (-3)\)[/tex]:
[tex]\(-2 \times -3 = 6\)[/tex], and the variable is [tex]\(x\)[/tex]
Result: [tex]\(6x\)[/tex]
3. [tex]\((-9y^2) \cdot (-4x)\)[/tex]:
[tex]\(-9 \times -4 = 36\)[/tex], and the variables [tex]\(y^2 \times x\)[/tex]
Result: [tex]\(36xy^2\)[/tex]
4. [tex]\((-9y^2) \cdot (-3)\)[/tex]:
[tex]\(-9 \times -3 = 27\)[/tex], and the variable is [tex]\(y^2\)[/tex]
Result: [tex]\(27y^2\)[/tex]
Step 3: Combine All the Terms
Now, combine all the individual results:
[tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex]
This matches one of the provided options:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the answer is:
[tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex]
