We can factor a polynomial by finding its roots. In particular, a quadratic equation has (at most) two roots [tex]x_1,\ x_2[/tex], which would allow us to write the polynomial as
[tex]p(x)=a(x-x_1)(x-x_2)[/tex]
To find the solutions, we can use the quadratic formula
[tex]x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} = \dfrac{-5\pm\sqrt{169}}{4} = \dfrac{-5\pm13}{4}[/tex]
So, the two solutions are
[tex]\dfrac{-5+13}{4} = 2\quad\text{and}\quad\dfrac{-5-13}{4} =-\dfrac{9}{2}[/tex]
And so we can factor the polynomial as follows:
[tex]2x^2+5x-18=2(x-2)\left(x+\dfrac{9}{2}\right)[/tex]
