As it appears the f(x) and g(x) will not be equal for any real value of x for the following reasons.
For in case x < 0 then f(x) = g(x) implies x + 3 = x + 1 which does not make any sense.
Now for x > = 0 we will get
x+3 = -x^2 + 2x + 1 that is x^2 -x + 2 = 0 and this quadratic does not have any real roots.