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We want to factor the expression $216+12x$ to find the dimensions of the extended stage.
First, let's factor out the greatest common factor of $216$ and $12$, which is $12$. We can rewrite the expression as: $$12(18+x)$$
Now, we have the expression $18+x$. In order to factor this further, we need to find two numbers whose product is $18$ and whose sum is $x$. The numbers $2$ and $9$ satisfy these conditions.
Therefore, we can factor the expression completely as: $$12(2+9)(9)$$
The dimensions of the extended stage are $2+9$ meters by $9$ meters, or $11$ meters by $9$ meters.