The standard equation of the quadratic function of vertex ( -5, -1 ) and passes through the point ( -2, 2)
[tex]\begin{gathered} a=\frac{1}{3} \\ \\ F(x)=\frac{1}{3}(x+5)^2+1 \\ f(x)=\frac{1}{3}(x^2+10x+25)+1 \\ \\ f(x)=\frac{1}{3}x^2+\frac{10}{3}x+\frac{22}{3} \\ \end{gathered}[/tex]So, the answer will be:
[tex]f(x)=\frac{1}{3}x^2+\frac{10}{3}x+\frac{22}{3}[/tex]