How do you completely factorize $5{{y}^{3}}-5{{y}^{2}}-10y$ ?

Hint: To completely factorize a polynomial expression, we will first find a common variable in all the individual terms. If there is a common variable then, we can take it out and factorize the remaining smaller degree polynomial. If there is no common variable in the expression then we use the hit and try method to find a root of our expression. We shall proceed like this to get the required answer to our problem.

Complete step by step solution:
We have been given the mathematical expression as: $5{{y}^{3}}-5{{y}^{2}}-10y$. Let us first check for a common variable term in all the individual terms or equations. If there isn’t any, then we will use the hit and try method.
In our expression, we can see that the variable ‘y’ is common in all the three terms. So, we don’t need the hit and try method. Our equation could be written as:
$=y\left( 5{{y}^{2}}-5y-10 \right)$
Here, the term inside the bracket is a quadratic. It can be factorized by breaking its mid-term into two terms. This can be done as follows:
$\begin{align}
  & =y\left( 5{{y}^{2}}-10y+5y-10 \right) \\
 & =y\left[ 5y\left( y-2 \right)+5\left( y-2 \right) \right] \\
 & =y\left( y-2 \right)\left( 5y+5 \right) \\
 & =5y\left( y-2 \right)\left( y+1 \right) \\
\end{align}$
Thus, the given mathematical expression has been completely factored.

Hence, $5{{y}^{3}}-5{{y}^{2}}-10y$ can be completely factored as $5y\left( y-2 \right)\left( y+1 \right)$.

Note:
The purpose of finding a common variable is that, when that variable is equated to zero, it acts as one of the roots of the expression which we try to find out by the hit and try method. It is like a short cut or trick to get one root of the expression without doing unnecessary and time taking calculations. The breaking of the middle term of our quadratic equation had the same purpose to yield the roots of quadratic.