By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $120$ and $q$ divides the leading coefficient $8$. One such root is $\frac{3}{2}$. Factor the polynomial by dividing it by $2x-3$.
Consider $4x^{3}-3x^{2}-42x-40$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-40$ and $q$ divides the leading coefficient $4$. One such root is $-2$. Factor the polynomial by dividing it by $x+2$.
$$\left(x+2\right)\left(4x^{2}-11x-20\right)$$
Consider $4x^{2}-11x-20$. Factor the expression by grouping. First, the expression needs to be rewritten as $4x^{2}+ax+bx-20$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-11$$ $$ab=4\left(-20\right)=-80$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-80$.
