1. The problem is to find the square of the sum of two variables $A$ and $b$, which is expressed as $(A + b)^2$. 2. The formula for the square of a binomial is given by: $$ (x + y)^2 = x^2 + 2xy + y^2 $$ 3. Applying this formula to $(A + b)^2$, we identify $x = A$ and $y = b$. 4. Substitute into the formula: $$ (A + b)^2 = A^2 + 2Ab + b^2 $$ 5. This means the square of the sum $A + b$ expands to the sum of the square of $A$, twice the product of $A$ and $b$, and the square of $b$. 6. Therefore, the final answer is: $$ (A + b)^2 = A^2 + 2Ab + b^2 $$