Question 750980
You would add a term and subtract the same term, allowing you to convert part of the expression into a factorable square trinomial.  Your given function would be handled this way:


{{{-x^2+4x-5=-1*(x^2-4x+5)}}}, factor a -1


See the non-square part which is {{{x^2-4x}}}, which can be factored into {{{x(x-4)}}}.  This is like a representation of a rectangle of x by (x-4) area.  You could imagine cutting the longer length  to leave a square area, cut the extra piece in half and put one of them along the other neighboring side of the square piece.  A drawing would be better, as would be found in a few intermediate algebra books.  Anyway, you will notice a missing square corner.  THAT represents the term to both add and subtract symbolically.

Continuing....

You want {{{(-4/2)^2=4}}}.  This is that missing square term.


{{{-1*(x^2-4x+5)=-1*(x^2-4x+4+5-4)}}}
={{{-1*((x-2)^2+5-4)}}}
={{{-1*((x-2)^2+1)}}}
={{{-1*(x-2)^2-1}}}, square completed.


The standard form for your function is {{{highlight(f(x)=-1(x-2)^2-1)}}}