Factor: -(x)(x - 4) > 0
The polynomial is even degree (2) with a negative leading coefficient (-1), meaning its graph begins in QIII, below the x-axis. (I.e. Its y-values are negative for negative x-values of large magnitude.) It has single roots at x = 0 and x = 4, which means the y-values change signs at both of those x-intercepts. Putting these facts together results in the following sign chart:
neg pos neg
<----------------|-----------------------------------|------------------->
0 4
Thus, the solution to the inequality is 0 < x < 4. In interval notation: x ∈ (0 , 4).
The sign chart above can be confirmed by graphing the polynomial and/or by interval testing (eg. plug in x = -1, x = 1, and x = 5).
