Consider the ideal $(4,x)$ in the ring $\mathbb{Z}[x]$. I want to calculate the radical of this ideal. Is my proposed calculation correct?
\begin{align*} r(4,x) &=& r(4) \cap r(x) \\ &=& r(2^2) \cap r(x) \\ &=& r(2) \cap r(x) \\ &=& r(2,x) \\ &=& (2,x). \end{align*}
As stated in my comment to your question, that derivation looks wrong.
Finding the radical of an ideal, given its generators, is not easy in general. However, in this case it's not difficult. First observe (obviously) that $2$ must be in the radical, and $X$ as well. So the radical contains $2$ and $X$. But the ideal $(2, X)$ is already a maximal ideal (because the quotient ring is $\Bbb Z / 2\Bbb Z$, a field), so it is not only contained in the radical of $(4, X)$, it must be equal to the radical.