Predict p

From the functional equation $f(f(x))+x^2+x+p=xf(x)$ , we note that the degree of the left hand side is at least $2$ . Let the degree of $f(x)$ be $n$ , then the degree of $f(f(x))$ is $2n$ . Since $2n\ge 2$ , the degree of the LHS is $2n$ . The degree of the RHS is $n+1$ . Equating the degrees on both sides, we have $2n=n+1\implies n =1$ . Then $f(x)$ is of the form $f(x) = ax+b$ , where $a$ and $b$ are constants. Then the functional equation becomes

$\begin{array} {rl} F(x): &a(ax+b)+b+x^2+x+p=x(ax+b) \\ F(x): &a^2x+ab+b+x^2+x+p = ax^2+bx \\ F(0): & ab+b+p = 0 \\ \implies F(x): &a^2x+x^2+x = ax^2+bx \\ F(1): & a^2+2 = a+b \\ F(-1): & -a^2 = a-b \\ F(1)+F(-1): & 2 = 2a \\ \implies & a = 1 \\ F(-1): & -1 = 1-b \\ \implies & b = 2 \\ \implies & f(x) = x + 2 \end{array}$

Then we have:

$\begin{aligned} f\left(2x+\frac 3x \right)&= \blue{2x+\frac 3x} + 2 & \small \blue{\text{By AM-GM inequality}} \\ & \ge \blue{2\sqrt 6} + 2 & \small \blue{\text{Equality occurs when }x=\sqrt{\frac 32}} \\ & \approx \boxed{6.899} \end{aligned}$