The line after 81 - 4k2 < 0 seems to be a step in the wrong direction.

4k2 > 81 is still not incorrect, but the following step is incorrect.

Correct is

4k2 - 81 > 0
(2k + 9) (2k - 9) > 0
k < -4.5 or k > 4.5

We take the case k < -4.5 as the coefficient of x2 must be negative for an expression to always be negative. Then, the largest corresponding possible value of k is -5.

Thanks for the amendments