First simplify the given equation:
#color(white)("XXX")y=color(orange)(-3x^2-2x-1)+color(brown)((2x-1)^2)#
#color(white)("XXX")y=color(orange)(-3x^2-2x-1)+color(brown)(4x^2-4x+1)#
#color(white)("XXX")y=x^2-6x#
One of the easiest ways of finding the vertex is to convert the equation into "vertex form":
#color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)# with vertex at #(color(red)(a),color(blue)(b))#
by "completing the square"
(Note that in this case we can ignore #color(green)(m)# or write it with its implied value of #color(green)(1)#).
#color(white)("XXXXXX")#Remember #(x+k)^2 = x^2+2kx+k^2#
#color(white)("XXXXXX")#So in this case #k=-3#
#color(white)("XXXXXX")# and we will need to add #(-3)^2# to complete the square
#color(white)("XXX")y=x^2-6xcolor(purple)(+9-9)#
#color(white)("XXX")y=(x-color(red)(3))^2+color(blue)("("-9")")#
which is in vertex form with the vertex at #(color(red)(3),color(blue)("("-9")"))#
Here is a graph of the original equation to help verify our result:
graph{-3x^2-2x-1+(2x-1)^2 [-7.46, 12.54, -10.88, -0.88]}
