Let's group the cubic function: [tex]h(x)=x^4+2x^3-10x^2-18x+9=(x-3)(x+3)(x^2+2x-1)[/tex].
The first two roots are [tex]\pm3[/tex] however to find the last two roots we need to solve the square equation:
[tex]x^2+2x-1=0
\\D=b^2-4ac=2^2-4\cdot1\cdot(-1)=8[/tex]. Now we know the discriminator, using the discriminator we're able to find the roots of the equation: [tex]x_{1,2}=\frac{-b\pm\sqrt{D}}{2a}=\frac{-2\pm\sqrt{8}}{2}=\frac{-2\pm2\sqrt{2}}{2}=\frac{2(\sqrt{2}\pm1)}{2}[/tex]. The roots of the square equation are [tex]x_{1}=-1-\sqrt{2}[/tex] and [tex]x_{2}=\sqrt{2}-1[/tex].
The roots of the cubic function are [tex]\pm3[/tex], [tex]-1-\sqrt{2}[/tex] and [tex]\sqrt{2}-1[/tex].
