Answer: $x = -\frac{224}{5}$.
Suppose that $x \ge 0$. Let us denote $t = [x] \ge 0$, and then we can write the following estimations:
$$t \le x < t+1 \Rightarrow t^2 \le x[x] < t^2 + t.$$
$44^2 + 44 = 1980 < 2016 < 45^2$, therefore there are no solutions of the equation among positive numbers.
Suppose that $x < 0$. Then denote $t = [x] < 0$, i.e. $-[x] = -t > 0$. Hence,
$$t \le x < t+1 \Rightarrow -t-1 \le -x < -t \Rightarrow t^2 + t \le x[x] \le t^2.$$
$44^2 = 1936 < 2016$ and $46^2 = 1936 < 2016 < 46^2 - 46$, therefore only possible value is $t = -45$. Indeed, $45^2 - 45 = 1980 < 2016 < 45^2 = 2025$, i.e. the solution is possible. Let us denote $x = -45 + y$, $0 \le y < 1$. Then it should hold that:
$$x[x] = -45 \cdot (-45 + y) = 2025 - 45y = 2016 \Rightarrow 45y = 9 \Rightarrow y = \frac{1}{5}.$$
Thus, the desired solution is $x = -45 + \frac{1}{5} = -\frac{224}{5}$.