What is the range of $λ$? Suppose $a, b, c$ are the sides of a triangle and no two of them are equal.
Let $λ ∈ IR$. If the roots of the equation $x^
2 + 2(a + b + c)x + 3λ(ab +
bc + ca) = 0$ are real, then what is the range of $λ$?
I got that $$λ ≤\frac{
(a + b + c)^
2}
{3(ab + bc + ca)}$$
After that what to do?

There is a systematic way to get rid of the triangle condition by substituting 
$$a = x + y, b = y + z, c = z + x$$
which comes from the tangents to the incircle, explained e.g. here. Now the only condition on $x,y,z$ is that they are positive reals, and additionally they are distinct (because we want $a,b,c$ to be distinct).
The quadratic has real roots iff the discriminant is $\geq 0$, and as you say, if (writing in terms of the new variables)
$$\lambda \leq \frac{4(x+y+z)^2}{3(x^2+y^2+z^2) + 9(xy+yz+zx)}$$
Now we need to find the range of that expression for distinct real $x,y,z$.
Now let $a=x^2+y^2+z^2$, $b=xy+yz+zx$, and observe that $a> b$ from $(x-y)^2+(y-z)^2+(z-x)^2>0$, and moreover $a/b$ can take any value $\alpha> 1$. Hence, since our expression is
$$\frac{4}{3}\frac{a+2b}{a+3b} = \frac{4}{3}\frac{a/b+2}{a/b+3}$$
and we know the range of $a/b$, we get the range of $\lambda$ as well.
