Both the numerator and denominator can be factorized, giving
((2x+1)(x+1))/((x-1)(x-4)).
Clearly this has two vertical asymptotes x=1 and x=4 (of the simple "1/x" type, and two x-axis intercepts at x=1/2 and x=-1. Putting x=0 gives the y-axis intercept as (0,1/4),
Since the numerator and denominator have highest powers of x which are the same, namely 2, there is a horizontal asympote y=2, the 2 being the ratio of the coefficients of those highest equal powers. (Informally, as x to oo, the two squares dominate over the other terms in the expression, so all the xs nearly cancel out, leaving the constant 2.)
Going from left to right, the graph must descend steadily from y=2 as x increases from -oo, then go through the two zeroes (and hence through some minimum between the zeroes), thence through the asympototic region at x=1, going through a maximum on its way to the second asymptotic region at x=4, thence decreasing steadily to the horizontal asymptote as x to oo.![enter image source here]()
