Since we have negative numbers under a square root operation, we obviously are in a domain of complex numbers.
Representing -32=16*2*(-1) and -8=4*2*(-1) we derive the following equivalent expression:
sqrt(-32)-sqrt(-8)=4*sqrt(2)*sqrt(-1)-2*sqrt(2)*sqrt(-1)
Now we have an interesting problem. The easy (but incomplete!) approach is replace sqrt(-1) with complex number i and derive
4*sqrt(2)*sqrt(-1)-2*sqrt(2)*sqrt(-1)=4*sqrt(2)*i-2*sqrt(2)*i=2*sqrt(2)*i.
It's incomplete because sqrt(-1)=+-i. Therefore, we have two values for each member:
4*sqrt(2)*sqrt(-1)=+-4*sqrt(2)i
2*sqrt(2)*sqrt(-1)=+-2*sqrt(2)i
That produces four different variants of the answer:
(a) sqrt(-32)-sqrt(-8)=4*sqrt(2)*i-2*sqrt(2)*i=2*sqrt(2)i
(b) sqrt(-32)-sqrt(-8)=4*sqrt(2)*i+2*sqrt(2)*i=6*sqrt(2)
(c) sqrt(-32)-sqrt(-8)=-4*sqrt(2)*i-2*sqrt(2)*i=-6*sqrt(2)
(d) sqrt(-32)-sqrt(-8)=-4*sqrt(2)*i+2*sqrt(2)*i=-2*sqrt(2)
All four answers are equivalent and represent a possible simplification of the original expression in normal complex form.
Now the question is, how is it possible that a single expression have 4 different representations in a normal complex form. The answer is simple. The operation of square root in the domain of complex numbers has two different values. Inasmuch as sqrt(-1) can be either i or -i (both, if squared, produce -1), any expression that contains sqrt(-1) has more than one representation in the complex form. We deal with an unusual type of a function - square root - that has two values for any single argument.
So, when we wright an expression sqrt(-32) or sqrt(-8), without additional assumption we cannot say what exactly it means, similarly to sqrt(-1). That's why we had multitude of expressions representing the original one in a normal complex form.
