What is the value of k so that the equation (k^2 + 2)x^2 - 3x^2 + 3x + k + 2 = 0 will have two distinct real roots?

If a quadratic equation ax^2 + bx + c = 0 has two distinct
real roots b^2 > 4ac

Here, the equation is (k^2 +
2)x^2 - 3x^2 + 3x + k + 2 = 0

=> x^2(k^2 + 2 - 3) +
3x + (k + 2) = 0

b^2 >
4ac

=> 9 > 4*(k^2 - 1)(k +
2)

=> 9 - 4*(k^2 - 1)(k + 2) >
0

=> 9 - 4*(k^3 - k + 2k^2 - 2) >
0

=> 9 - 4k^3 + 4k - 8k^2 + 8 >
0

=> -4k^3 - 8k^2 + 4k + 17 >
0

=> 4k^3 + 8k^2 - 4k - 17 <
0

I don't see how this can be solved further to yield
actual values of k, especially as the expression has complex
roots.

For the equation to have 2 distinct
real roots, the value of k should satisfy : 4k^3 + 8k^2 -
4k - 16 < 0.