The number of distinct real roots of the equation $$x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0$$ is ______________.

The number of real solutions of the equation $${e^{4x}} + 4{e^{3x}} - 58{e^{2x}} + 4{e^x} + 1 = 0$$ is ___________.

Let $$\alpha$$, $$\beta$$ be the roots of the equation $${x^2} - 4\lambda x + 5 = 0$$ and $$\alpha$$, $$\gamma$$ be the roots of the equation $${x^2} - \left( {3\sqrt 2 + 2\sqrt 3 } \right)x + 7 + 3\lambda \sqrt 3 = 0$$, $$\lambda$$ > 0. If $$\beta + \gamma = 3\sqrt 2 $$, then $${(\alpha + 2\beta + \gamma )^2}$$ is equal to __________.

If the sum of all the roots of the equation

$${e^{2x}} - 11{e^x} - 45{e^{ - x}} + {{81} \over 2} = 0$$ is $${\log _e}p$$, then p is equal to ____________.