To find the product of [tex]\(2 x^4\left(2 x^2 + 3 x + 4\right)\)[/tex], we will follow the distributive property to expand the polynomial expression.
[tex]\[
2 x^4 \left(2 x^2 + 3 x + 4\right)
\][/tex]
We need to distribute [tex]\(2 x^4\)[/tex] to each term within the parenthesis:
[tex]\[
= 2 x^4 \cdot 2 x^2 + 2 x^4 \cdot 3 x + 2 x^4 \cdot 4
\][/tex]
Let's expand each term separately:
1. [tex]\(2 x^4 \cdot 2 x^2\)[/tex]:
[tex]\[
2 \cdot 2 \cdot x^4 \cdot x^2 = 4 x^{4+2} = 4 x^6
\][/tex]
2. [tex]\(2 x^4 \cdot 3 x\)[/tex]:
[tex]\[
2 \cdot 3 \cdot x^4 \cdot x = 6 x^{4+1} = 6 x^5
\][/tex]
3. [tex]\(2 x^4 \cdot 4\)[/tex]:
[tex]\[
2 \cdot 4 \cdot x^4 = 8 x^4
\][/tex]
Now, combining all these terms together, the expanded polynomial is:
[tex]\[
4 x^6 + 6 x^5 + 8 x^4
\][/tex]
Therefore, the product of [tex]\(2 x^4 \left(2 x^2 + 3 x + 4\right)\)[/tex] is:
[tex]\[\boxed{4 x^6 + 6 x^5 + 8 x^4}\][/tex]
