Divide through by -2..
\(\displaystyle -2x^2 + 11x - 5 = 0 \Rightarrow x^2 - \frac{11}{2} x + \frac{5}{2} = 0 \\\)
Now complete the square:
1) divide -11/2 by 2 you get -11/4
2) square result from number 1.. 121/16
3) Add and subtract 121/16 from the original equation:
\(\displaystyle x^2 - \frac{11}{2} x + \frac{5}{2} + \frac{121}{16} - \frac{121}{16} = 0\)
4) As a direct result, \(\displaystyle x^2 - \frac{11}{2} x + \frac{121}{16}\) factors into: \(\displaystyle (x - \frac{11}{4})^2\)
5) So, \(\displaystyle x^2 - \frac{11}{2} x + \frac{5}{2} + \frac{121}{16} - \frac{121}{16} = (x - \frac{11}{4})^2 + \frac{5}{2} - \frac{121}{16} = (x - \frac{11}{4})^2 + \frac{40}{16} - \frac{121}{16} = (x - \frac{11}{4})^2 - \frac{81}{16} = 0 \\\)
6) Finally, given \(\displaystyle (x^2 - \frac{11}{4})^2 - \frac{81}{16} = 0\) solve for x:
\(\displaystyle (x - \frac{11}{4})^2 = \frac{81}{16}\)
\(\displaystyle x - \frac{11}{4} = \, +/- \, \sqrt{\frac{81}{16}}\)
\(\displaystyle x = \frac{11}{4} \, +/- \, \frac{\sqrt{81}}{4}\)
\(\displaystyle x = \frac{11\,\, +/- \,\, 9}{4}\)
x = 5 or x = 1/2
You could also use the quadratic equation, which in many cases will be easier.