Factorize: \[mn + m + n + 1\]

Hint:
Here we need to factorize the given algebraic expression. For that, we will first regroup the terms and then we will find the factors which are common. We will take out all the common factors from all the terms. This will give us the required factorization of the given expression.

Complete step by step solution:
Here we factorize \[mn + m + n + 1\] using regrouping technique.
Now, we will group the first two terms and the last two terms of the expression.
\[mn + m + n + 1 = \left( {mn + m} \right) + \left( {n + 1} \right)\]
Now, we will take out the common term \[m\] from the first group. Therefore, we get
\[ \Rightarrow mn + m + n + 1 = m\left( {n + 1} \right) + \left( {n + 1} \right)\]
Now, we can see that both the grouped terms are containing a common factor i.e. \[\left( {n + 1} \right)\].
So, we will take out the common factor \[\left( {n + 1} \right)\] from these two groups to get the factors.
\[ \Rightarrow mn + m + n + 1 = \left( {m + 1} \right)\left( {n + 1} \right)\]
We have got two factors of the given algebraic expression \[\left( {m + 1} \right)\] and \[\left( {n + 1} \right)\].
Thus, the factorization of the given algebraic expression is \[\left( {m + 1} \right)\left( {n + 1} \right)\].

Hence, this is the required expression.

Note:
Here, factorization of algebraic expression means to split the given algebraic expression into their factors. Factors of the algebraic expression are the terms which when used to divide the given algebraic expression will give us the remainder zero. Sometimes, all the terms of the expression do not have a common factor. But these terms can be grouped in such a way that all the terms in each group can have a common factor. When we do this common factor comes out from all the groups and will give us the required factorization of the expression.