Answer :
To find the product of [tex]\(2x^4\)[/tex] and [tex]\((2x^2 + 3x + 4)\)[/tex], we'll follow these steps:
1. Apply the Distributive Property:
Multiply [tex]\(2x^4\)[/tex] with each term inside the parentheses:
[tex]\[
2x^4 \cdot (2x^2 + 3x + 4) = (2x^4 \cdot 2x^2) + (2x^4 \cdot 3x) + (2x^4 \cdot 4)
\][/tex]
2. Multiply Each Term:
- First Term: [tex]\(2x^4 \cdot 2x^2\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 2 = 4\)[/tex].
- Add the exponents of [tex]\(x\)[/tex]: [tex]\(4 + 2 = 6\)[/tex].
- Result: [tex]\(4x^6\)[/tex].
- Second Term: [tex]\(2x^4 \cdot 3x\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 3 = 6\)[/tex].
- Add the exponents of [tex]\(x\)[/tex]: [tex]\(4 + 1 = 5\)[/tex].
- Result: [tex]\(6x^5\)[/tex].
- Third Term: [tex]\(2x^4 \cdot 4\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 4 = 8\)[/tex].
- The power of [tex]\(x\)[/tex] remains 4, as it is multiplied by a constant.
- Result: [tex]\(8x^4\)[/tex].
3. Combine All Terms:
Combine the results from each multiplication:
[tex]\[
4x^6 + 6x^5 + 8x^4
\][/tex]
This final polynomial [tex]\(4x^6 + 6x^5 + 8x^4\)[/tex] represents the expanded form of the given expression. Therefore, the correct answer from the options is:
- [tex]\(4x^6 + 6x^5 + 8x^4\)[/tex]
1. Apply the Distributive Property:
Multiply [tex]\(2x^4\)[/tex] with each term inside the parentheses:
[tex]\[
2x^4 \cdot (2x^2 + 3x + 4) = (2x^4 \cdot 2x^2) + (2x^4 \cdot 3x) + (2x^4 \cdot 4)
\][/tex]
2. Multiply Each Term:
- First Term: [tex]\(2x^4 \cdot 2x^2\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 2 = 4\)[/tex].
- Add the exponents of [tex]\(x\)[/tex]: [tex]\(4 + 2 = 6\)[/tex].
- Result: [tex]\(4x^6\)[/tex].
- Second Term: [tex]\(2x^4 \cdot 3x\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 3 = 6\)[/tex].
- Add the exponents of [tex]\(x\)[/tex]: [tex]\(4 + 1 = 5\)[/tex].
- Result: [tex]\(6x^5\)[/tex].
- Third Term: [tex]\(2x^4 \cdot 4\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 4 = 8\)[/tex].
- The power of [tex]\(x\)[/tex] remains 4, as it is multiplied by a constant.
- Result: [tex]\(8x^4\)[/tex].
3. Combine All Terms:
Combine the results from each multiplication:
[tex]\[
4x^6 + 6x^5 + 8x^4
\][/tex]
This final polynomial [tex]\(4x^6 + 6x^5 + 8x^4\)[/tex] represents the expanded form of the given expression. Therefore, the correct answer from the options is:
- [tex]\(4x^6 + 6x^5 + 8x^4\)[/tex]
