How do I determine if there are common factors for x^5+3x^3+5x/4X^4+2x^2, when the leading coefficient of the denominator is greater than that of the numerator, and there is no constant in the numerator - thus making using the roots calculator impossible, and long division.

I need to identify if any the vertical, horizontal and oblique asymptotes.

f(x) = (x^5+3x^3+5x) / (4x^4+2x^2)

The only common factor of numerator and denominator is x, but it isn't really necessary to cancel it.

i. Vertical Asymptote
To get this, we get the limit of x, as f(x) approaches infinity.
lim x as f(x)->infinity
Or easier way is to get the values of x where the denominator is zero. Thus we equate the denominator to zero:
4x^4 + 2x^2 = 0
(2x^2)(2x^2 + 1) = 0
The only real value of x is equal to 0. The other two roots are imaginary.
Thus the vertical asymptote is x = 0

ii. Horizontal Asymptote
To get this, we get the limit of f(x), as x approaches infinity.
lim f(x) as x->infinity
lim (x^5+3x^3+5x) / (4x^4+2x^2)
Note that, if the highest degree of x in the numerator is greater than the highest degree of x in the denominator, there is no horizontal asymptote, but there is an oblique asymptote.

iii. Oblique Asymptote
To get the oblique asymptote, perform long division and get the quotient part (exclude the remainder). This equation is the oblique asymptote.
Long division cannot be shown here, but you should get y = x/4.

Hope this helps~ :3