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?

For a triangle with sides $a,b,c$ by triangle inequality, we have 
$$|a-b|<c$$ 
Squaring both sides we get,
$$(a-b)^2<c^2\tag{1}$$
Similarly, 
$$(b-c)^2<a^2\tag{2}$$
And
$$(c-a)^2<b^2\tag{3}$$ 
Adding $(1),(2)$ and $(3)$, we get
$$a^2+b^2+c^2 <2(ab+bc+ca) \Longleftrightarrow (a+b+c)^2 <4(ab+bc+ca)\tag{4}$$
From $(4)$, we get $\lambda <\dfrac{4}{3}$

To see why the upper bound is the supremum consider a degenerate triangle with sides $(a,a,c)$,(where $c \rightarrow 0$) and permutations.
For example, when $(a,b,c)=(1,1, 0.01)$, then $\lambda \leq \dfrac{4.0401}{3.0600} \approx 1.321 $
