There are many ways of solving this equation, which is called a second order polynomial (where the order describes the highest power of x), let's focus on the most general for now: the quadratic formula.
For any equation of the form ax2+bx+c=0, we can use this formula to find values of x for which this equation is satisfied. Start by calculating the discriminant of our equation, given by D = b2-4ac. If D is bigger than 0, than we can apply the quadratic formula. In our example, a=1, b=3 and c=-4, which means D=32-(-4*4)=9+16=25, which is bigger than 1, so it works!
The values of x can then be found using the quadratic formula,
x1=(-b+(squareroot(D))/(2a)
x2=(-b-(squareroot(D))/(2a)
Notice the difference between those two formulas is the + and - in front of the square root of D.
For our example, this would give:
x1=(-3+(squareroot(25))/(2*1)=(-3+5)/2=1
x2=(-3-(squareroot(25))/(2*1)=(-3-5)/2=-4
You can check that this is correct by replacing x with one of your values and verify is does equal 0!