Since \(\displaystyle (x-2)\) is a factor of \(\displaystyle f(x) = 5x^3+12x^2-23x-42\), then \(\displaystyle 5x^3+12x^2-23x-42 = (x-2)q(x)\), for some polynomial \(\displaystyle q(x)\). Divide \(\displaystyle f(x)\) by \(\displaystyle (x-2) \) to find \(\displaystyle q(x)\), and then factor the quadratic equation that you find. You will get \(\displaystyle (x-2)(x-\alpha)(x-\beta)\) as the required factors, where \(\displaystyle \alpha\) and \(\displaystyle \beta\) are the roots of the quadratic equation \(\displaystyle q(x) = 0\).