(x2+4x)/(4x+4).
DOMAIN
The denominator is zero when x=-1, so the domain is all values of x except x=-1 (asymptote).
Domain is (-∞,-1), (-1,∞).
RANGE
(-∞,∞)
DEGREE
The function is not a polynomial but its degree is 1, because the denominator divides into the numerator to produce a quotient in which the highest power of x is 1.
INTERCEPTS
When y=0, x2+4x=0, x(x+4)=0 so the x intercepts are 0 and -4.
When x=0, y=0, so the y-intercept is 0.
ASYMPTOTES
Vertical asymptote is x=-1. There is no horizontal asymptote but there is a sloping asymptote y=¼x. This is found by ignoring all but the highest power of x in the denominator and numerator:
If X is a very large value, (X2+4X)/(4X+4)≈X2/4X=X/4.
HOLES
None. There are no discontinuities apart from the asymptotes.