Investigate the behaviour of the function \[ f(x) = x^4+4x^3-2x^2-12x+5, \] and determine the roots of the equation \(f(x)=0\) accurately.
Solution: \begin{align*} && f(x) &= x^4+4x^3-2x^2-12x+5 \\ &&&= (x(x+2))^2 - 6x^2 - 12x + 5 \\ &&&= (x(x+2))^2 - 6(x(x+2)) + 5 \\ &&&= y^2 - 6y + 5 \tag{\(y = x^2+2x\)} \\ &&&=(y-5)(y-1) \\ &&&= (x^2+2x-5)(x^2+2x-1) \\ && f(x) &= 0 \\ \Rightarrow && 0 &= ((x+1)^2-6)((x+1)^2 - 2)) \\ \Rightarrow && x &= -1 \pm \sqrt{6}, -1 \pm \sqrt{2} \end{align*}