Factorize the expression $abx^2+(ay-b)x-y$.

Given:

The given expression is $abx^2+(ay-b)x-y$.

To do:

We have to factorize the expression $abx^2+(ay-b)x-y$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. 

An algebraic expression is factored completely when it is written as a product of prime factors.

Here, we can factorize the expression $abx^2+(ay-b)x-y$ by grouping similar terms and taking out the common factors.

We can write $abx^2+(ay-b)x-y$ as,

$abx^2+(ay-b)x-y=abx^2+axy-bx-y$

The terms in the given expression are $abx^2, ayx, -bx$ and $-y$.

We can group the given terms as $abx^2, -bx$ and $axy, -y$. 

Therefore, by taking $bx$ as common in $abx^2, -bx$ and $y$ as common in $axy, -y$, we get,

$abx^2+(ay-b)x-y=bx(ax-1)+y(ax-1)$

Now, taking $(ax-1)$ common, we get,

$abx^2+(ay-b)x-y=(bx+y)(ax-1)$

Hence, the given expression can be factorized as $(bx+y)(ax-1)$.