y = -2x² - 12x - 10

Factor -2 out of only the first two terms on the right

    y = -2(x² + 6x) - 10

Complete the square inside the parentheses:
1. Multiply the coefficient of x, which is +6 times 1/2, getting +3
2. Square +3 getting +9
3. Add and subtract 9 inside the parentheses  

    y = -2(x² + 6x + 9 - 9) - 10

Change the parentheses to brackets and group the first 
three terms inside the bracket in parentheses:

    y = -2[(x² + 6x + 9) - 9] - 10

Factor the trinomial inside the parentheses as a perfect square
That is, go straight to it or else x²+6x+9 = (x+3)(x+3) = (x+3)²

    y = -2[(x + 3)² - 9] - 10

Remove the bracket by distributing, leaving the (x+3)² intact:

    y = -2(x + 3)² + 18 - 10

Combine the constant terms on the right

    y = -2(x + 3)² + 8

Add -8 to both sides:

y - 8 = -2(x + 3)²

Edwin