ANSWER
[tex]y = - 3( {x + 3)}^{2} - 4[/tex]
EXPLANATION
The given function is
[tex]y = - 3 {x}^{2} - 18x - 31[/tex]
Factor -3
[tex]y = - 3( {x}^{2} + 6x) - 31[/tex]
Add and subtract the square of half the coefficient of x.
[tex]y = - 3( {x}^{2} + 6x + {3}^{2} ) - - 3( {3)}^{2} - 31[/tex]
[tex]y = - 3( {x}^{2} + 6x + {3}^{2} ) + 27 - 31[/tex]
Apply perfect squares,
[tex]y = - 3( {x +3)}^{2} - 4[/tex]
The vertex form is:
[tex]y = - 3( {x +3)}^{2} - 4[/tex]
