For a equation of the form:
[tex]y=ax^2+bx+c[/tex]The vertex form is given by:
[tex]y=a(x-h)^2+k[/tex]Where:
[tex]\begin{gathered} h=-\frac{b}{2a} \\ k=y(h) \end{gathered}[/tex]so:
For:
[tex]\begin{gathered} y=-5x^2-15x-31 \\ a=-5 \\ b=-15 \\ c=-31 \end{gathered}[/tex][tex]\begin{gathered} h=-\frac{(-15)}{2(-5)}=-\frac{3}{2}=-1.5 \\ k=-5(-1.5)^2-15(-1.5)-31=-\frac{79}{4}=-19.75 \end{gathered}[/tex]Therefore, the vertex form of the equation is:
[tex]\begin{gathered} y=-5(x+\frac{3}{2})^2-\frac{79}{4} \\ or \\ y=-5(x+1.5)^2-19.75 \end{gathered}[/tex]