f(x)=sqrt(x^2+a*x+4)-3

3 = (X^2 + Ax + 4)^(1/2)
SQUARE BOTH SIDES
9 = X^2 + Ax + 4
TRANSFER ALL TERMS TO ONE SIDE
0 = X^2 + Ax + 4 - 9
0 = X^2 + Ax - 5
0 = (X+1)(X-5)
0 = X^2 -5x + x - 5
0 = X^2 -4x - 5
A = -4

it can be positive or negative 4 depending on the signs