How to factor cubic polynomials? My polynomial: $2x^3 +7x^2+12x+9$.
Now, I've tried both of the techniques given in this Wikihow page, but neither of them worked for this problem. Synthetic division is something which I think would work, but it seems like a lot of work to first use the Rational Zeros Theorem to check all the possibilities using synthetic division and then come up with a quadratic equation.
The solution given is as follows: solution in Slader.
Obviously, one cannot use synthetic division to factor our $(2x +3)$ for the division works only for factors of the form $(x-c)$. So my question is, how do you factor a polynomial like this?

The polynom $2x^3 +7x^2+12x+9$ is a polynomial with coefficients in $\mathbb{Q}$, there is a result saying that the roots living in $\mathbb{Q}$ are of the form $\frac{a}{b}$ where $a$ divides thecoefficient $a_0$ and $b$ divides the dominant coefficient of the polynomial.
So in our case : $$a\in \{ -9,-3,-1,1,3,9\}$$
$$b\in \{-2,-1,1,2\}$$ which can be reduced to $$b\in \{1,2\}$$ because otherwise each fraction appears twice.
Since all the coefficient are positive, a positive root will automatically give a quantity $> 0$. So :$$a\in \{ -9,-3,-1\}$$
$$b\in \{2,1\}$$
From here you can try the $6$ possible rationnal roots.
