Factorize the following expression, \[8{{\left( a-2b \right)}^{2}}-2a+4b-1\] .

Hint: The given expression is \[8{{\left( a-2b \right)}^{2}}-2a+4b-1\] . Now, take -2 as common from the term \[\left( -2a+4b \right)\] and simplify the given expression. Then, take the term \[\left( a-2b \right)\] as common from the simplified expression. Now, simplify it further and get the required answer.

Complete step by step answer:
According to the question, we have an expression given and we have to factorize it, that is, we have to simplify the given expression.
The given expression = \[8{{\left( a-2b \right)}^{2}}-2a+4b-1\] ………………………………………….(1)
We can observe that the given expression has the term, \[\left( -2a+4b \right)\] .
From equation (1), we have the given expression.
Now, on taking -2 as common from the term \[\left( -2a+4b \right)\] from the given expression, we get
\[=8{{\left( a-2b \right)}^{2}}-2a+4b-1\]
\[=8{{\left( a-2b \right)}^{2}}-2\left( a-2b \right)-1\] ……………………………………………(2)
Here, we can observe that in equation (2), we have the term \[\left( a-2b \right)\] that is repeated twice.
Now, on taking the term \[\left( a-2b \right)\] as common in equation (2) and simplifying it, we get
\[=8{{\left( a-2b \right)}^{2}}-2\left( a-2b \right)-1\] ……………………………………(3)
In equation (3), we can see that our given expression has been simplified more.
We can also observe that the term \[\left( a-2b \right)\] can be taken as common.
Since the term \[8{{\left( a-2b \right)}^{2}}\] has the square of the term \[\left( a-2b \right)\] so, on taking the term \[\left( a-2b \right)\] as common we will get one term as \[8\left( a-2b \right)\] in the bracket.
Now, on taking the term \[\left( a-2b \right)\] as common in equation (3), we get
\[=\left( a-2b \right)\left\{ 8\left( a-2b \right)-2 \right\}-1\] ………………………………………..(4)
Now, from equation (4), we have the most simplified form of the given expression and we know that factorization of an expression is its simplified form.
Therefore, the factorization of the expression \[8{{\left( a-2b \right)}^{2}}-2a+4b-1\] is \[\left( a-2b \right)\left\{ 8\left( a-2b \right)-2 \right\}-1\] .

Note:
In this question, one might do a silly mistake while taking the term \[\left( a-2b \right)\] as common and then write the expression as \[\left( a-2b \right)\left[ \left\{ 8\left( a-2b \right)-2 \right\}-1 \right]\] . This is wrong because when we expand the expression, we get \[\left[ \left\{ 8{{\left( a-2b \right)}^{2}}-2\left( a-2b \right) \right\}-1\left( a-2b \right) \right]\] which is not equal to the original expression. Therefore, be careful while taking the term \[\left( a-2b \right)\] as common.