The number of real solutions of

$${x^7} + 5{x^3} + 3x + 1 = 0$$ is equal to ____________.

The number of distinct real roots of x⁴ $$-$$ 4x + 1 = 0 is :

Let a, b $$\in$$ R be such that the equation $$a{x^2} - 2bx + 15 = 0$$ has a repeated root $$\alpha$$. If $$\alpha$$ and $$\beta$$ are the roots of the equation $${x^2} - 2bx + 21 = 0$$, then $${\alpha ^2} + {\beta ^2}$$ is equal to :

The sum of all the real roots of the equation

$$({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$$ is