Polynomials over the complex numbers still have the unique factorization property, so if a polynomial has a real factorization, that will also be its complex factorization. This also means that you can go about factorizing in the same way you would over the reals. The ways you already know for quadratics (quadratic formula, completing the square, etc.) still work fine, except now you keep complex answers instead of throwing them out.
To check your answer, just multiply the factors back together, using complex multiplication. In this case, your answer isn't right, since $(1.5-3.25i)(1.5+3.25i)=1.25^2+3.25^2=12.8125$. In fact, this polynomial has real roots, which I'm sure you can find if you look at $6(x^2+3x-10)$ for a while.
(As a reminder, a quadratic has real roots if and only if its discriminant is nonnegative.)
EDIT: appears somebody else got to the answer first.
EDIT 2 in light of your most recent edit:
I don't think you really understand completing the square. That's actually okay, since you can do everything with the quadratic formula, but at your level in school you'll probably get tested on that specific technique, so you might as well learn it. Also, it's how you derive the quadratic formula in the first place.
As far as I can see, you made three mistakes:
First, you had $x^2-3x-10$ instead of $x^2+3x-10$.
Second, you forgot some parentheses when you completed the square.
Third, "taking the square root" didn't really work out. Here, it's better to work with the equation $f(x)=0$ than just the expression $f(x)$. Then, you can move the $12.25$ to the other side of the equation and take the square root, giving $\pm 3.5$, not $\pm 3.5i$. (Also, you changed $3.5i$ to $3.25i$ on the way back.) When you add it back in, this gives you the expected answer, $6(x-5)(x+2)$.
That isn't the answer to the equation the first way you wrote it, but I can't tell which is the right equation.
I guess the moral is that you can check your work at nearly any step of these types of problems. If you do this frequently, you can avoid little mistakes like these. Good luck!