To determine if the polynomial \(2x^2 - kx - 12\) is divisible by \(x - 4\), we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial \(P(x)\) is divisible by \(x - a\), then \(P(a) = 0\).

In this case, we have \(P(x) = 2x^2 - kx - 12\) and \(a = 4\). So, we need to find the value of \(k\) for which \(P(4) = 0\).

Substitute \(x = 4\) into the polynomial:

\(P(4) = 2(4)^2 - k(4) - 12\)

\(P(4) = 32 - 4k - 12\)

\(P(4) = 20 - 4k\)

For the polynomial to be divisible by \(x - 4\), \(P(4)\) must equal 0. Therefore:

\(20 - 4k = 0\)

\(4k = 20\)

\(k = \frac{20}{4}\)

\(k = 5\)

So, the value of \(k\) that makes \(2x^2 - kx - 12\) divisible by \(x - 4\) is k = 5.