Validating a search string for an SQL text search
Laravel's Eloquent uses PDO's parameterized queries, so you get the same level of protection from SQL injection attacks as using PDO directly. Which is to say, you don't generally need to worry about sanitizing your input with Laravel.

The main exception being if you ever use Eloquent's raw query. You should avoid passing user-supplied input to a raw query at pretty much any price, but that's where you'd need to sanitize.

As for the input validation, you should just be using it to make sure the inputs conform to your requirements. For instance, ensuring that something looks like an email or is alphanumeric. Or to make sure inputs correspond to an extant record in the database. It's almost never necessary to use it for input sanitization and will probably end up confusing both you and the user, later on.