If ${x^2} + px + 1$ is a factor of the expression $a{x^3} + bx + c$, then

The number of non-negative integer solutions of the equations $6 x+4 y+z=200$ and $x+y+z=100$ is

The number of real roots of the equation $x | x |-5| x +2|+6=0$, is

The number of distinct real roots of the equation $|\mathrm{x}+1||\mathrm{x}+3|-4|\mathrm{x}+2|+5=0$, is ...........

Let $a$ be the largest real root and $b$ be the smallest real root of the polynomial equation $x^6-6 x^5+15 x^4-20 x^3+15 x^2-6 x+1=0$ Then $\frac{a^2+b^2}{a+b+1}$ is

If the sum of the two roots of the equation $4{x^3} + 16{x^2} - 9x - 36 = 0$ is zero, then the roots are