Roots of a quadratic equation with a parameter

Solution : For $m\in ]-\infty,0[\cup ]0,1[$.

There is a trap : one is tempted to use the discriminant, which is :
$$\Delta=4(m+1)^2-4 m(m+3)=4\left[\left(m^2+2 m+1\right)-\left(m^2+3 m\right)\right]=4(1-m)$$so the discriminant is strictly positive if and only if $m<1$.
However, the discriminant can be only used when the quadratic equation is not degenerate, that is, when $m\neq 0$. For $m=0$, the equation is linear, with only one solution. So $m=0$ has to be removed from the admissible $m$.

This is a great illustration of why it is important to make sure we fully understand every term in a problem before trying to solve it, since this can change the solution.