This is a link to a step by step guide to my shortcut approach. When applied properly it should only take about 4 to 5 lines all depending on the complexity of the question. https://socratic.org/s/aMg2gXQm

The objective is to have the format #y=a(x+b/(2a))^2+c+k#

Where #k# is a correction making #y=a(x+b/(2a))^2+c color(white)("d")# have the same overall values as #y=ax^2+bx+c#

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#color(blue)("Answering the question - the more formal approach")#

#color(brown)("This is one of those situations where you just have to")##color(brown)("remember the standard form steps")#

Lets multiply out the brackets

#y=-1/7( 6x-3)(x/3+5) " "..................Equation(1)#

# y=-1/7(2x^2+30x-x-15)#

#y=-1/7(2x^2+29x-15)#

Factor out the 2 from #2x^2#. We do not want any coefficient in front of the #x^2#

# y=-2/7(x^2+29/2x-15/2) #

Just for ease of reference set #g=x^2+29/2x-15/2# giving:

#y=-2/7g" "..........................Equation(1_a)#

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The x-intercepts are at #y=0# giving

#y_("x-intercept")=0=-2/7g#

Thus it must be true that for this condition #g=0# thus we have:

#g=0=x^2+29/2x-15/2#

Add #15/2# to both sides

#15/2=x^2color(red)(+29/2)x#

To make the right hand side into a perfect square we need to add #(1/2xxcolor(red)(29/2))^2->(29/4)^2# so add #841/16# to both sides giving:

#15/2+ 841/16color(white)("d")=color(white)("d")x^2+29/2x+841/16#

#15/2+ 841/16color(white)("d")=color(white)("d")(x+29/4)^2#

#961/16color(white)("d")=color(white)("d")(x+29/4)^2#

#g=0=(x+29/4)^2-961/16#

But from #Equation(1_a)" "y=-2/7g# giving

#y=0=-2/7[(x+29/4)^2-961/16]#

#color(magenta)("I will let you finish this off.")#