Q1
\(\displaystyle (x-1)(x-2)>0\)
If (x-1)<0 and (x-2)<0, then f(x)>0, as (-)(-)=(+), hence x<1
If (x-1)>0 and (x-2)>0, then f(x)>0, as (+)(+)=(+), hence x>2
\(\displaystyle -\infty<x<1\)
\(\displaystyle 2<x<\infty\)
Q2
\(\displaystyle x^2+x+1>0\ is\ of\ the\ form\ ax^2+bx+c>0\ and\ doesn't\ factorise\ with\ integers.\)
\(\displaystyle x=0\ at\ x_1\ and\ x_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-1\pm\sqrt{1-4}}{2}=\frac{-1\pm\sqrt{-3}}{2}\)
Complex roots indicate that this graph never crosses the x-axis.
f(0)=1, therefore the graph is entirely above the x axis.
f(x)>0 for all x from -infinity to infinity.
