To find the product of the given expression [tex]\( x^3 \left( x^2 + 5x + 1 \right) \)[/tex], let's break it down into detailed steps:
1. Start with the expression:
[tex]\[
x^3 \left( x^2 + 5x + 1 \right)
\][/tex]
2. Distribute [tex]\( x^3 \)[/tex] to each term inside the parentheses:
- Multiply [tex]\( x^3 \)[/tex] by [tex]\( x^2 \)[/tex]:
[tex]\[
x^3 \cdot x^2 = x^{3+2} = x^5
\][/tex]
- Multiply [tex]\( x^3 \)[/tex] by [tex]\( 5x \)[/tex]:
[tex]\[
x^3 \cdot 5x = 5 \cdot x^{3+1} = 5x^4
\][/tex]
- Multiply [tex]\( x^3 \)[/tex] by [tex]\( 1 \)[/tex]:
[tex]\[
x^3 \cdot 1 = x^3
\][/tex]
3. Combine all the results from the multiplication:
[tex]\[
x^5 + 5x^4 + x^3
\][/tex]
So, the product of [tex]\( x^3 \left( x^2 + 5x + 1 \right) \)[/tex] is:
[tex]\[
x^5 + 5x^4 + x^3
\][/tex]
Thus, the correct answer is:
[tex]\[
x^5 + 5x^4 + x^3
\][/tex]
