Functional Equation

$P(P(x)) = 6x - P(x)\\ P(P(x)) + P(x) = 6x$

Now, let the degree of $P(x)$ be $n$ . The degree of $P(P(x))$ is $n^n$

For the sum of a $n$ th degree polynomial and a $n^n$ th degree polynomial to be a polynomial of degree $1$ , the only possible value of $n$ is $1$ .

Therefore, $P(x)$ is a polynomial of the form $ax+b$ .

Substitute it into the equation:

$a(ax+b) + b + ax + b = 6x\\ a^2x + ax + ab + 2b = 6x\\ (a^2 + a)x + ab + 2b = 6x$

By comparison:

$(a^2 + a)x = 6x \quad\quad ab+2b = 0$

Solve for $a$ :

$(a^2 + a)x = 6x\\ a^2+a = 6\\ a^2 + a - 6 = 0\\ (a+3)(a-2) = 0\\ a=2,\;-3$

When $a=2$ :

$2b+2b = 0\\ 4b=0 \implies b=0$

When $a=-3$ :

$-3b+2b = 0\\ -b=0 \implies b=0$

There are two possible expressions of $P(x)$ , and they are:

$P(x) = 2x \implies P(10) = 2(10) = 20\\ P(x) = -3x \implies P(10) = -3(10) = -30$

The product of all possible values of $P(10)$

$=20 \times -30 = \boxed{-600}$