Answer :
Sure! Let's use the distributive property to expand [tex]\(5x^4(-4x^3 - 7x^2 + 2x + 1)\)[/tex]. We'll distribute [tex]\(5x^4\)[/tex] to each term inside the parenthesis:
1. Distribute [tex]\(5x^4\)[/tex] to [tex]\(-4x^3\)[/tex]:
[tex]\[
5x^4 \times -4x^3 = -20x^{4+3} = -20x^7
\][/tex]
2. Distribute [tex]\(5x^4\)[/tex] to [tex]\(-7x^2\)[/tex]:
[tex]\[
5x^4 \times -7x^2 = -35x^{4+2} = -35x^6
\][/tex]
3. Distribute [tex]\(5x^4\)[/tex] to [tex]\(2x\)[/tex]:
[tex]\[
5x^4 \times 2x = 10x^{4+1} = 10x^5
\][/tex]
4. Distribute [tex]\(5x^4\)[/tex] to [tex]\(1\)[/tex]:
[tex]\[
5x^4 \times 1 = 5x^4
\][/tex]
Now, let's combine all these expanded terms into one polynomial expression:
[tex]\[
-20x^7 - 35x^6 + 10x^5 + 5x^4
\][/tex]
So, the expanded form of the expression using the distributive property is:
[tex]\[
-20x^7 - 35x^6 + 10x^5 + 5x^4
\][/tex]
This matches the second option listed in the original question.
1. Distribute [tex]\(5x^4\)[/tex] to [tex]\(-4x^3\)[/tex]:
[tex]\[
5x^4 \times -4x^3 = -20x^{4+3} = -20x^7
\][/tex]
2. Distribute [tex]\(5x^4\)[/tex] to [tex]\(-7x^2\)[/tex]:
[tex]\[
5x^4 \times -7x^2 = -35x^{4+2} = -35x^6
\][/tex]
3. Distribute [tex]\(5x^4\)[/tex] to [tex]\(2x\)[/tex]:
[tex]\[
5x^4 \times 2x = 10x^{4+1} = 10x^5
\][/tex]
4. Distribute [tex]\(5x^4\)[/tex] to [tex]\(1\)[/tex]:
[tex]\[
5x^4 \times 1 = 5x^4
\][/tex]
Now, let's combine all these expanded terms into one polynomial expression:
[tex]\[
-20x^7 - 35x^6 + 10x^5 + 5x^4
\][/tex]
So, the expanded form of the expression using the distributive property is:
[tex]\[
-20x^7 - 35x^6 + 10x^5 + 5x^4
\][/tex]
This matches the second option listed in the original question.
