1. **State the problem:** Show that $ (\sin A + \cos A)^2 = 1 + \sin 2A $ and find the maximum value of $ 4(\sin A + \cos A)^2 $. 2. **Expand the left side:** $$ (\sin A + \cos A)^2 = \sin^2 A + 2 \sin A \cos A + \cos^2 A $$ 3. **Use the Pythagorean identity:** $$ \sin^2 A + \cos^2 A = 1 $$ 4. **Substitute the identity:** $$ (\sin A + \cos A)^2 = 1 + 2 \sin A \cos A $$ 5. **Use the double-angle identity for sine:** $$ \sin 2A = 2 \sin A \cos A $$ 6. **Rewrite the expression:** $$ (\sin A + \cos A)^2 = 1 + \sin 2A $$ 7. **Hence, the expression is shown.** 8. **Find the maximum value of $4(\sin A + \cos A)^2$:** $$ 4(\sin A + \cos A)^2 = 4(1 + \sin 2A) $$ 9. **Since $\sin 2A$ ranges from $-1$ to $1$, the maximum value of $1 + \sin 2A$ is $1 + 1 = 2$.** 10. **Therefore, the maximum value is:** $$ 4 \times 2 = 8 $$ **Final answers:** $$ (\sin A + \cos A)^2 = 1 + \sin 2A $$ Maximum value of $4(\sin A + \cos A)^2$ is $8$.