Does anyone know if and how this 6x^2 - 15x - 21 can be solved with the “completing the square technique” from prepswift?

Any quadratic can be solved using any valid technique - it depends on how *useful* that is.

In this case, certainly:

6(x^{2} - \frac{5}{2} x - \frac{7}{2}) - by taking 6 out.

You should now be able to complete the square of x^2 - \frac{5}{2}x - \frac{7}{2}.

The result would be 6(x - \frac{5}{4})^2 + k, where k is a real I leave you to find.