convert $2 x^{2}-3 x-2$ to vertex form

Step 1 ：Factor out the coefficient of the x^2 term: \(y = 2(x^2 - \frac{3}{2}x) - 2\)

Step 2 ：Complete the square by adding and subtracting the square of half of the coefficient of the x term: \(y = 2\left(x^2 - \frac{3}{2}x + \left(\frac{-3}{4}\right)^2 - \left(\frac{-3}{4}\right)^2\right) - 2\)

Step 3 ：Rewrite the equation in vertex form: \(y = 2\left(x - \frac{3}{4}\right)^2 - 2 + 2\left(\frac{-3}{4}\right)^2\)

Step 4 ：Calculate the final equation: \(y = 2\left(x - \frac{3}{4}\right)^2 - \frac{7}{8}\)

Step 5 ：\(\boxed{y = 2\left(x - \frac{3}{4}\right)^2 - \frac{7}{8}}\)