Given $a,b,c$ positive real values, $\lambda$ any real and the polynomial $p(x)= x^2+x(a+b+c)+\lambda(ab+ bc+ac)$. I have to prove that if $\lambda=\frac{3}{4}$ then the roots of $p(x)$ are real. I have been thinking about the polynomial of degree 3 given by the roots $a,b,c$, however, I am not finding a way through.
The problem has another question but I feel like I have to solve the first one to underestand this: Prove that if $a,b,c$ are the lenghts of a triangle and $\lambda \geq 1$ then $p(x)$ does not have real roots.
EDIT: I have tried some ideas of the commnets and I have proved easily the first part knowing that $(a^2+b^2+c^2)-(ab+cb+ac)\geq 0$. For the second part it was a direct proof knowing the Ravi's substitution, something unknown for me until now.