There are two ways, by completing the square:

First way:

    y = 2x² - x - 1

Get the constant term on the left by adding 1 to both sides:

y + 1 = 2x² - x

To get the first term on the right to just x² by multiplying
every term through by 

y +  = x² - x

Complete the square on the right side of the equation:
1.  Multiply the coefficient of x which is  by , getting 
2.  Square that amount  = 

3. Add that amount  to both sides of the equation:

y +  + = x² - x + 


Factor the right side as a perfect square:

y +  + = (x - )²

Clear of fractions by multiplying every term on both sides by 16

8y + 8 + 1 = 16(x - )²

Combine 8 + 1 as 9

8y + 9 = 16(x - )

Subtract 9 from both sides

8y = 16(x - )² - 9

Solve for y by dividing every term by 8

 y = 2(x - )² - 

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Second way:

    y = 2x² - x - 1

Factor 2 out of the first two terms:

    y = 2(x² - x) - 1

Change the parentheses to brackets so they can hold parentheses:

    y = 2[x² - x] - 1

Complete the square in the parentheses:
1.  Multiply the coefficient of x which is  by , getting 
2.  Square that amount  = 

3. Add and subtract that amount  inside the brackets:

    y = 2[x² - x +  - ] - 1

Factor the first three terms in the bracket as a perfect square:

    y = 2[(x - )² - ] - 1

Remove the bracket by distributing the 2, leaving the parentheses intact:

    y = 2(x - )² -  - 1

    y = 2(x - )² -  - 1

Combine the last two terms, by writing 1 as 

    y = 2(x - )² -  - 

    y = 2(x - )² - 

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Take your pick.

Edwin