It's essential that you know the quadratic formula; you'll see it repeatedly.
In -2x^2 - 3x + 2 = 0, a= -2, b= -3 and c= 2.
Let's first find the discriminant, so that we can know whether the roots are real or complex.
The discriminant is b^2 - 4(a)(c), or (-3)^2 - 4(-2)(2). This comes out to 9 + 16, or 25. Thus we know that the solutions are real, not complex.
sqrt(25) = plus or minus 5.
Thus, the roots are:
-(-3) plus or minus 5 3+5 3-5
x = -------------------------------- = --------- = -2 and ------------ = 1/2
2(-2) -4 -4