Factorize the algebraic expression $ \left( {10ab + 4a + 5b + 2} \right) $

Hint: In the given question, we are provided with an algebraic expression. We have to factorise the algebraic expression given in the question using algebraic rules and techniques. Algebraic expressions can be factored by taking out common terms outside of the bracket and simplifying the rest of the terms inside the bracket.

Complete step by step solution:
For factorising the given algebraic expression $ \left( {10ab + 4a + 5b + 2} \right) $ , we can use the fcatorisation method in which we take the common factors out of the bracket and simplify the rest of the terms.
So, we have, $ \left( {10ab + 4a + 5b + 2} \right) $
Firstly, we group the terms in such a way that there is some common factor in the terms of each group. So, we get,
 $ \Rightarrow $ $ \left( {10ab + 4a} \right) + \left( {5b + 2} \right) $
Now, we take out the common factor $ 2a $ outside of the first bracket,
 $ \Rightarrow $ $ 2a\left( {5b + 2} \right) + \left( {5b + 2} \right) $
Now, we have the bracket $ \left( {5b + 2} \right) $ common in both groups. So, taking the bracket common, we get,
 $ \Rightarrow $ $ \left( {5b + 2} \right)\left( {2a + 1} \right) $
So, factored form of the algebraic expression $ \left( {10ab + 4a + 5b + 2} \right) $ is $ \left( {5b + 2} \right)\left( {2a + 1} \right) $ .
So, the correct answer is “ $ \left( {5b + 2} \right)\left( {2a + 1} \right) $ .”.

Note: Such problems revolve around the concepts of factorisation and simplification. One should have a strong grip over algebraic and simplification rules such as taking common factors outside the bracket and transposition rule.