Explain what value or values of c make the equation x^2+12x+c=0 have 1 and only one solution ( justify the answer) so

Only one solution means thtat it factors to a perfect square

to do that, we do
the number in front of the x^2 is 1
then take 1/2 of the number in front of the x term and square it and that is the constant term
12/2=6 6^2=36

36=c

x^2+12x+36=0
(x+6)^2=0
x+6=0
x=-6