Answer
There are 8 electrons surrounding the $Si$ atom.
Work Step by Step
In the formation of $SiCl_4$ molecule, with each $Cl$ atom sharing 1 electron, the $Si$ atom also shares 1 electron.
Therefore, with 4 $Cl$ atoms for 1 $Si$ atom, the $Si$ atom would share all 4 of its valence electrons and get back 8 valence electrons that can be used by all the atoms.
Therefore, surrounding the $Si$ atom is 4 sharing spots, with each spot having 2 electrons. That means surrounding the $Si$ atom is 8 valence electrons, which is an octet.
In the diagram, we saw that in the $SiCl_4$ molecule, there are 4 lines and no dots surrounding the $Si$ atom. Each line represents a shared electron pair, which means there are 8 electrons surrounding the $Si$ atom.
