Let us show by contradiction that x5+10x3+x+1 is not O(x4). Suppose that x5+10x3+x+1 is O(x4). Then there exist a constant C and x0 such that x5+10x3+x+1≤Cx4 for all x≥x0. It follows that x4x5+10x3+x+1≤C for all x≥x0.
Since x→∞limx4x5+10x3+x+1=x→∞lim(x+x10+x31+x41)=∞, we conclude that there exists x1>x0 such that x4x15+10x13+x1+1>C. This contradiction proves that x5+10x3+x+1 is not O(x4).