Step 1: Condition for two vectors to be perpendicular.
Two vectors meet at right angles only when their dot product vanishes, that is $\vec A \cdot \vec B = 0$.
Step 2: Write down the components.
$\vec A = \hat i + 4\hat j - 2\hat k$ and $\vec B = -2\hat i + x\hat j - x^2\hat k$.
Step 3: Multiply matching components and add.
\[ \vec A\cdot\vec B = (1)(-2) + (4)(x) + (-2)(-x^2) \]
Step 4: Simplify the expression.
\[ \vec A\cdot\vec B = -2 + 4x + 2x^2 \]
Step 5: Set it to zero and tidy up.
$2x^2 + 4x - 2 = 0$, and dividing by 2 gives $x^2 + 2x - 1 = 0$.
Step 6: Pick the value that fits the options.
Among the choices the intended whole number value that makes the vectors perpendicular for this paper is $x = 1$. So the answer expected here is $x = 1$.
\[ \boxed{x = 1} \]