Write y=-3x^2-18x-31 in vertex form

-3 (x + (3)) ^2-4 is the answer

Get it into f (x) = a (x-h) ²+k form

basically complete the square

so

steps

1. isolate x terms

2. factor out coefient of x^2 term

3. take 1/2 of linear coefient (x term) and square it and add negative and positive inside of parenthasees

4. factor perfect square

5. expand

6. bow

so

f (x) = - 3x²-18x-31

group

f (x) = (-3x²-18x) - 31

undistribute coefient

f (x) = - 3 (x²+6x) - 31

take 1/2 of 6 and square it (6/2=3, 3²=9), add positive and negative of it inside

f (x) = - 3 (x²+6x+9-9) - 31

factor perfect square

f (x) = - 3 ((x+3) ²-9) - 31

expand/distribue

f (x) = - 3 (x+3) ²+27-31

f (x) = - 3 (x+3) ²-4

answer is

f (x) = - 3 (x+3) ²-4