Vertical asymptotes exist when the denominator would equal zero since you cannot divide by zero.
x^4 +3 = 0
x^4 = -3
This statement is never true because a variable raised to an even root is always positive. This means that there are no vertical asymptotes.
Horizontal aysmptopes are determined by the following rules. If the highest power in the top is greater than the highest power in the bottom, there are none. If the powers are even (like in this case) then an aysmptote exists at y = division of the leading coefficients. In this case there is an asymtope at y = 4/1 = 4. If the higher power is in the bottom, then the asymptote is along the x axis.