HOW TO "ACTUALLY" SPLIT AN INFINITIVE...

The notion that the split infinitive is grammatically wrong is based on traditonal grammar books, which took their rules from Latin.
 A SPLIT INFINITIVE is just a grammatical construction in which a word or phrase, usually an adverb, comes between  TO and the bare infinitive of a verb.

See if these sentences could be expressed in a different way without changing the meaning:

 

YOU HAVE TO RELUCTANTLY DO YOUR HOMEWORK

I PROMISE TO NEVER PLAY ANYTHING ORIGINAL

SHE´S GOT TO REALLY WATCH OUT FOR THIEVES

I NEED TO IMMEDIATELY VISIT THE LOO

I AM GOING TO THOROUGHLY WASH THE DOG

THIS WAS MEANT TO BETTER UNDERSTAND THE POEM

WE WENT TO AYO´S TO ENJOYABLY EAT PAELLA

I AM GOING TO TOTALLY PULVERISE HIM

WE ARE DETERMINED TO UTTERLY ERADICATE THE DISEASE

HE WANTS TO GRADUALLY GET RID OF PREJUDICES

THE POPULATION IS EXPECTED TO MORE THAN DOUBLE IN THE NEXT TEN YEARS

WE ALL WANT TO BOLDLY SPLIT INFINITIVES AT WILL

(famous Star Trek quotation) " TO BOLDLY GO WHERE NO MAN HAS GONE BEFORE"