$$b\left( a+1 \right)-a\left( b-1 \right)=ab+b-ab+a=b+a$$, so the answer is$\text{A}$$. 
Simplify: $$b\left( a+1 \right)-a\left( b-1 \right)$$． 
  - $$b+a$$ 
  - $$2ab+b-a$$ 
  - $$b-a$$ 
  - $$ab-1$$ 