1. The problem asks why $a^2 + b^2$ is different from $(a + b)^2$. 2. The formula for the square of a sum is: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ This means when you square a sum, you get the sum of the squares plus twice the product of the two terms. 3. On the other hand, $a^2 + b^2$ is simply the sum of the squares of $a$ and $b$ without the middle term $2ab$. 4. To see the difference clearly, expand $(a + b)^2$: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ Compare this to: $$ a^2 + b^2 $$ 5. The key difference is the $2ab$ term, which accounts for the interaction between $a$ and $b$ when squared together. 6. Therefore, $a^2 + b^2$ is not equal to $(a + b)^2$ unless $ab = 0$ (meaning either $a$ or $b$ is zero). This explains why the two expressions are different.