Posted by Misty on Thursday, September 15, 2011 at 11:39am.
Method 1: use a graphing tool and enter the function.
Method 2: Using several values for x, evaluate the function and plot the points (x,f(x)) to see the shape of the curve.
If you can't evaluate f(x), then you might reconsider taking pre-calculus.
As a starting point, note that f = -x^2(x^2+4), so you know there is a double root at x=0, so the graph is tangent to the x-axis at (0,0). Since it's a quartic, as x gets large, f gets large negative. Since there are no other real roots, the graph may look kind of like a flattened parabola opening downward. That will be the case here if f(x) < 0 for x near 0. Indeed, f(x) < 0 near x=0, so we know the graph is tangent to the x-axis from below.
FYI: Quartics may also look like a rounded W shape, But since the graph opens down, and is tangent below the x-axis, there can't be any bumps nearby, because then the graph would cross the x-axis near x=0. That would mean there are real roots.