To simplify the expression [tex]\(2 x^2\left(-4 x^4 + 6 x^3 - 2 x^2\right)\)[/tex], let's distribute [tex]\(2 x^2\)[/tex] through the terms inside the parentheses step by step.
1. Start with the given expression:
[tex]\[
2 x^2 (-4 x^4 + 6 x^3 - 2 x^2)
\][/tex]
2. Distribute [tex]\(2 x^2\)[/tex] to each term inside the parentheses:
- First term: [tex]\(2 x^2 \cdot (-4 x^4)\)[/tex]
[tex]\[
2 x^2 \cdot (-4 x^4) = -8 x^{2 + 4} = -8 x^6
\][/tex]
- Second term: [tex]\(2 x^2 \cdot 6 x^3\)[/tex]
[tex]\[
2 x^2 \cdot 6 x^3 = 12 x^{2 + 3} = 12 x^5
\][/tex]
- Third term: [tex]\(2 x^2 \cdot (-2 x^2)\)[/tex]
[tex]\[
2 x^2 \cdot (-2 x^2) = -4 x^{2 + 2} = -4 x^4
\][/tex]
3. Combine the results of these individual multiplications:
[tex]\[
-8 x^6 + 12 x^5 - 4 x^4
\][/tex]
Therefore, the simplified form of the expression [tex]\(2 x^2 (-4 x^4 + 6 x^3 - 2 x^2)\)[/tex] is:
[tex]\[
-8 x^6 + 12 x^5 - 4 x^4
\][/tex]
Among the given options, this result matches:
[tex]\[
-8 x^6 + 12 x^5 - 4 x^4
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{-8 x^6 + 12 x^5 - 4 x^4}
\][/tex]
