I'm not sure how represent radicals, but here is the problem:

-(sq root)-76 + (sq root)-125

It is my understanding that negative numbers do not have real-number square roots, so does that mean that this whole expression does not exist as a real number or what? Please help!
Thank you!

It looks like you have not learned about imaginary numbers or complex numbers

by definition √-1 = i

so -√-76 + √-125
= -√36√2√-1 + √25√5√-1
= -6√2 i + 5√5 i

yes, the answer does not exist in the set of real numbers.