Factorize the expression $ab(x^2+1)+x(a^2+b^2)$.

Given:

The given expression is $ab(x^2+1)+x(a^2+b^2)$.

To do:

We have to factorize the expression $ab(x^2+1)+x(a^2+b^2)$.

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 $ab(x^2+1)+x(a^2+b^2)$ by grouping similar terms and taking out the common factors. 

$ab(x^2+1)+x(a^2+b^2)$ can be written as,

$ab(x^2+1)+x(a^2+b^2)=ab(x^2)+ab(1)+x(a^2)+x(b^2)$

$ab(x^2+1)+x(a^2+b^2)=abx^2+ab+a^2x+b^2x$

The terms in the given expression are $abx^2, ab, a^2x$ and $b^2x$.

We can group the given terms as $abx^2, a^2x$ and $ab, b^2x$. 

Therefore, by taking $ax$ as common in $abx^2, a^2x$ and $b$ as common in $ab, b^2x$, we get,

$abx^2+ab+a^2x+b^2x=ax(bx+a)+b(a+bx)$

$abx^2+ab+a^2x+b^2x=ax(a+bx)+b(a+bx)$

Now, taking $(a+bx)$ common, we get,

$abx^2+ab+a^2x+b^2x=(ax+b)(a+bx)$

Hence, the given expression can be factorized as $(ax+b)(a+bx)$.