By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $60$ and $q$ divides the leading coefficient $15$. One such root is $1$. Factor the polynomial by dividing it by $x-1$.
Consider $15x^{3}+109x^{2}+104x-60$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-60$ and $q$ divides the leading coefficient $15$. One such root is $\frac{2}{5}$. Factor the polynomial by dividing it by $5x-2$.
$$\left(5x-2\right)\left(3x^{2}+23x+30\right)$$
Consider $3x^{2}+23x+30$. Factor the expression by grouping. First, the expression needs to be rewritten as $3x^{2}+ax+bx+30$. To find $a$ and $b$, set up a system to be solved.
$$a+b=23$$ $$ab=3\times 30=90$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. List all such integer pairs that give product $90$.
