Plug \(x = t\) and \(y = 4\) in \(y = 2x^2 - 2\)
\(4 = 2t^2 - 2\)
\(2t^2 = 6\)
\(t^2 = 3\)
\(t = ±\sqrt{3}\)

Now, Plug \(x = p\) and \(y = 0\) in \(y = 2x^2 - 2\)
\(0 = 2p^2 - 2\)
\(2p^2 = 2\)
\(p^2 = 1\)
\(p = ±1\)

(Refer to the figure below)
Greatest possible value of slope would be given when the line k passes through (\(\sqrt{3}\), 4) and (1, 0)

m = \(\frac{(4 - 0)}{(\sqrt{3} - 1)} = \frac{4}{0.732} = 5.46\)