use a graph, or guess and then confirm a bracket, or tabulate f(x) on a wide interval to figure the bracket of length one. to find an interval of length 1 that contains a root of the equation 3x^3 +x^2=x+5.
I don't understand what to do
consider the table for the two functions shown
x f(x) g(x)
-5 -350 0
-4 -176 1
-3 -72 2
-2 -20 3
-1 -2 4
0 0 5
1 4 6
2 28 7
3 90 8
4 208 9
5 400 10
It is clear that f(x)=g(x) somewhere in [1,2], since f(x)<g(x) when x<1 and f(x)>g(x) when x>2.
Or, if you rearrange things so that
f(x) = 3x^3+x^2-x-5 = 0
then just look for the interval where f(x) changes from negative to positive. You will fins that it is somewhere in [1,2].
The graphs are at
http://www.wolframalpha.com/input/?i=solve+3x^3+%2Bx^2%3Dx%2B5