regular expression to accept all strings other than one containing ads or adlib
Let $[$^$x]$, $[$^$xy]$ be short cuts for $\Sigma - \{x\}$, $\Sigma - \{x, y\}$ etc., i.e., $[$^$a]$ will be regular expression: $(b+c+d+\dots)$.

Then the regular expresseion for words not containing ads or adlib would be

$[$^$a]^*(a[$^$d][$^$a]^*+ad([$^$sl][$^$a]^* $ $+ adl[$^$i][$^$a]^* + adli[$^$b][$^$a]^*)^* (\epsilon + a +ad + adl + adli)$.

Basically we wait for an $a$ and then check it is not followed by the disallowed combinations.

Since regular expressions have built in operators for union, kleene and concatenation only, we will always have trouble with intersection, complement and minus operations. Basically the operation you want in your question is the complement.