Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$3y^{2}-2y=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $y=\frac{2±2}{6}$ when $±$ is plus. Add $2$ to $2$.
$$y=\frac{4}{6}$$
Reduce the fraction $\frac{4}{6}$ to lowest terms by extracting and canceling out $2$.
$$y=\frac{2}{3}$$
Now solve the equation $y=\frac{2±2}{6}$ when $±$ is minus. Subtract $2$ from $2$.
$$y=\frac{0}{6}$$
Divide $0$ by $6$.
$$y=0$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{2}{3}$ for $x_{1}$ and $0$ for $x_{2}$.
$$3y^{2}-2y=3\left(y-\frac{2}{3}\right)y$$
Subtract $\frac{2}{3}$ from $y$ by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
$$3y^{2}-2y=3\times \frac{3y-2}{3}y$$
Cancel out $3$, the greatest common factor in $3$ and $3$.
