Proving roots will be real If the roots of the equation $x^2+bx+c=0$ are real.
 Show that the roots of the equation $x^2+bx+c(x+a)(2x+b)=0$
are again, real for every real number $a.$

Not true.
Take $c=-1$. Then the discriminant of $x^2+bx+c$ is $b^2 + 4 > 0$ and so the roots are real.
The discriminant of $x^2+bx+c(x+a)(2x+b)$ is $4 a (a - b)$, which is negative if $0 < a < b$.
Therefore, the roots of $x^2+bx+c(x+a)(2x+b)$ are not real when $a=b/2$ and $b>0$, for instance. Indeed, in this case, the discriminant is $-b^2 < 0$.
