Factorize the given expression and find the most simplified expression possible after factorization, the given expression is : \[{a^2} + {b^2} - a + b\]

Hint: The given question need to solve by taking common the factors, which are possible to solve, and rest we have to write the equation as it is already, and on taking common if some similarities occur then more factors can be formed accordingly and can be simplified further.

Complete step by step solution:
The given question is \[{a^2} + {b^2} - a + b\]
Here we need to solve it by taking common the similar terms, and finally when the terms should be analyzed for similarity and then further simplification can be done, and can be solved further accordingly, on solving we get:
\[
   \Rightarrow {a^2} + {b^2} - a + b \\
   \Rightarrow a(a - 1) + b(b - 1) \\
 \]
The given question can not be simplified further, because no other common factors are now arising in the expression, hence it is the final simplified expression after factorizing the given expression in the question.

Additional Information:
Here we stopped after taking common only one time, because after that no possible term is there which can be factorized further, hence only this is the most possible factorized equation of the given expression in the question above.

Note: The given question is asked to factorize the given expression which needs to be factorized by taking common of the similar factors, and then after simplification we can solve further and get the final simplified and factorized equation. No other method is used until and unless the given expression is a multiple of a known expression.