(3x5+x4-17x3-4x2+20x)/(x3-4x) simplifies to:
(3x4+x3-17x2-4x+20)/(x2-4) because x is a common factor.
3x2+x-5 (quotient)
x2-4 ) 3x4+x3-17x2-4x+20
-(3x4 -12x2)
x3 -5x2-4x
-(x3 -4x)
-5x2 +20
-(-5x2 +20)
0 (no remainder)
CHECK
(x2-4)(3x2+x-5)=3x4+x3-5x2-12x2-4x+20=3x4+x3-17x2-4x+20.