get into form a(x-h)²+k
complete the square
-3x²-18x-31
group x terms
(-3x²-18x)-31
undistribute the -3
-3(x²+6x)-31
take 1/2 of the linear coefient (6) and squaer it
6/2=3, 3²=9
add positive and negative of it inside parenthasees
-3(x²+6x+9-9)-31
complete the square
-3((x+3)²-9)-31
expand
-3(x+3)²+27-31
-3(x+3)²-4 is vertex form