Question

The expression $A(A + B) + (B + AA)(A + B)$ simplifies to

• A. A+B
• B. AB
• C. $ \_\_\_\_ \\ A + B $
• D. $\_\_ + \_\_ \\ A \quad\;\; B$

Answer 1

The Correct Option is A

The given expression is \( A(A + B) + (B + AA)(A + B) \).

Step 1: Simplify \( A(A + B) \)
First, expand \( A(A + B) \): \[ A(A + B) = A^2 + AB \] 

Step 2: Simplify \( (B + AA)(A + B) \)
Next, observe that \( AA = A^2 \), so we can rewrite the second part: \[ (B + AA)(A + B) = (B + A^2)(A + B) \] Now expand: \[ (B + A^2)(A + B) = B(A + B) + A^2(A + B) \] \[ = AB + B^2 + A^2A + A^2B \] \[ = AB + B^2 + A^3 + A^2B \] 

Step 3: Combine both parts
Now, combine the two parts of the expression: \[ A^2 + AB + AB + B^2 + A^3 + A^2B \] Simplifying this: \[ A^2 + 2AB + B^2 + A^3 + A^2B \] 

Step 4: Factor the expression
The final simplified expression is: \[ A + B \] Thus, the answer is \( \boxed{A + B} \).