Question 1209234
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Write three first terms in this form

    2ab - a + b + ___ = 2*(ab - a/2 + b/2 + ___).


Look at the expression in parentheses.  

It is clear that it should be

    (ab - a/2 + b/2 - 1/4),


which is the product of linear binomials (a+1/2)*(b-1/2).


Now you have this identity

    2ab - a + b - 1/2  = 2*(a+1/2)*(b-1/2).


You can relate the factor 2 to the first or to the second binomial factor.  
It will give you two possible decompositions

    2ab - a + b - 1/2 = (2a+1)*(b-1/2)

or 

    2ab - a + b - 1/2 = (a+1/2)*(2b-1).


In any case, the blank term is -1/2.
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Solved.