If the rooots of the equation $x^2+bx+c=0$ are real , show that the roots of the equation $x^2 +bx+c(x+a)(2x+b)$ are again real for every real number a.
I assumed the discriminant of the first eqution to be $\geq 0$ and found a relation between the coefficients. I wrote down the discriminant of the second equation and using the relation from the first equation , I tried to show that this discriminant also must be $\geq 0$, but I landed up with complicated calculations . Since the two equation look similar to each other , I believe there is a smarter way of doing this problem.
The result is false when $a=2$, $b=4$, $c=-1$.