The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is ______.

Given,

$x\left(x^2+3\left|x\right|+5|x-1|+6|x-2|\right)=0$

$\Rightarrow x=0 \text{or} x^2+3\left|x\right|+5|x-1|+6|x-2|=0$

We know that sum of all positive terms can never be zero,

So, $x^2+3\left|x\right|+5|x-1|+6|x-2|$ can never be zero for any value of $x$,

So there is no value of $x$ which satisfies the above equation
Hence, only solution is $x=0$, i.e. number of solution is $1$ .