By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-2$ and $q$ divides the leading coefficient $12$. One such root is $\frac{2}{3}$. Factor the polynomial by dividing it by $3x-2$.
$$\left(3x-2\right)\left(4x^{2}-5x+1\right)$$
Consider $4x^{2}-5x+1$. Factor the expression by grouping. First, the expression needs to be rewritten as $4x^{2}+ax+bx+1$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-5$$ $$ab=4\times 1=4$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $4$.
$$-1,-4$$ $$-2,-2$$
Calculate the sum for each pair.
$$-1-4=-5$$ $$-2-2=-4$$
The solution is the pair that gives sum $-5$.
$$a=-4$$ $$b=-1$$
Rewrite $4x^{2}-5x+1$ as $\left(4x^{2}-4x\right)+\left(-x+1\right)$.
$$\left(4x^{2}-4x\right)+\left(-x+1\right)$$
Factor out $4x$ in the first and $-1$ in the second group.
$$4x\left(x-1\right)-\left(x-1\right)$$
Factor out common term $x-1$ by using distributive property.
