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