First,
#(f - g)(x) = f(x) - g(x) = (16 - x^2) - g(x)#
We can now write this equation and solve for #g(x)#:
#-x^2 + x + 12 = (16 - x^2) - g(x)#
First, add #color(red)(g(x))# to each wide of the equation:
#color(red)(g(x)) - x^2 + x + 12 = (16 - x^2) - g(x) + color(red)(g(x))#
#g(x) - x^2 + x + 12 = (16 - x^2) - 0#
#g(x) - x^2 + x + 12 = 16 - x^2#
Now add #color(red)(x^2)# and subtract #color(blue)(x)# and #color(green)(12)# from each side of the equation to solve for #g(x)# while keeping the equation balanced:
#g(x) - x^2 + color(red)(x^2) + x - color(blue)(x) + 12 - color(green)(12) = 16 - color(green)(12) - x^2 + color(red)(x^2) - color(blue)(x)#
#g(x) - 0 + 0 + 0 = 4 - 0 - x#
#g(x) = 4 - x#
