I was wondering the same the other day, what I found on internet was that the best way to make a search with like and text is 'WORD%' with the % at the end (text that end with word will not be found) if you can't use that your query will go row by row and word by word, reading all the text of each result to find the matches.
Other Options that I saw (it doesn't make too much sense to me) it's to separate the query,
select * from actor where first_name like 'alex%' or first_name like '%alex';
I think there is a full text index that will help a lot (let me see if I can find the info and edit my post)
EDIT: here it's the information of full text index,
first, engine HAVE TO BE MyISAM
add the index text,
ALTER TABLE fulltext_sample ADD FULLTEXT(column_name)
A list of the main features of a standard FULLTEXT search follows:
Excludes partial words
Excludes words less than 4 characters in length (3 or less)
Excludes words that appear in more than half the rows (meaning at least 3 rows are required)
Hyphenated words are treated as two words
Rows are returned in order of relevance, descending
Words in the stopword list (common words) are also excluded from the search results. The stopword list is based upon common English words, so if your data is used for a different purpose, you'll probably want to change the list. Unfortunately, doing so at present is not easy. You'll need to edit the file myisam/ft_static.c. recompile MySQL, and rebuild the indexes! To save you hunting through the source, or if you have a binary version of MySQL, here is a list of stopwords. Note that these can and do change with different versions. To be absolutely sure, you'll have to check the specific list for your version.
With this you can improve your search by fulltext using the '%Axel%'