Let's simplify the Boolean expression step by step using the rules of Boolean algebra.

The expression is:

$A'(a + b) + (b + aa)(a + b')$

Step 1: Apply the Idempotent Law (which states $A + A = A$) to simplify $aa$ in the second term.

$A'(a + b) + (b + a)(a + b')$ This is because $aa = a$.

Step 2: Expand both terms.

1. First term: $A'(a + b)$

[ A'(a + b) = A'a + A'b