Determine whether the statement is true or false. If the \[ \begin{aligned} a^{2}+b^{2} & =a^{2}+a b-a b+b^{2} \\ & =a(a+b)-b(a+b) \\ & =(a+b)(a-b) \end{aligned} \] Choose the correct answer below. A. False. The correct statement is as follows \[ \begin{aligned} a^{2}-b^{2} & =a^{2}+a^{2} b-a b^{2}-b^{2} \\ & -a^{2}(a+b)-b^{2}(a+b) \\ & =\left(a^{2}-b^{2}\right)(a+b) \end{aligned} \] B. False. The correct statement is as follows. \[ \begin{aligned} a^{2}-b^{2} & =a^{2}+a b-a b-b^{2} \\ & =a(a+b)-b(a+b) \\ & =(a+b)(a-b) \end{aligned} \] c. False. The correct statement is as tollows \[ \begin{aligned} a^{2}+b^{2} & =a^{2}-a b+a b-b^{2} \\ & -a(a-b)+b(a-b) \\ & =(a+b)(a-b) \end{aligned} \] D. True

The original statement claims that: \[ \begin{aligned} a^{2} + b^{2} &= a^{2} + a b -a b + b^{2} \\ &= a(a + b) - b(a + b) \\ &= (a + b)(a - b) \end{aligned} \] Let's verify its validity: 1. **Simplifying \((a + b)(a - b)\):** \[ (a + b)(a - b) = a(a - b) + b(a - b) = a^{2} - ab + ab - b^{2} = a^{2} - b^{2} \] 2. **Comparing to the original left-hand side:** \[ a^{2} + b^{2} \neq a^{2} - b^{2} \] unless \( b = 0 \), which is not generally true. Therefore, the original statement is **false**. **Option B** correctly identifies the mistake and provides the correct statement: \[ \begin{aligned} a^{2} - b^{2} &= a^{2} + a b - a b - b^{2} \\ &= a(a + b) - b(a + b) \\ &= (a + b)(a - b) \end{aligned} \] This is the standard factorization for the difference of squares. **Answer:** **B. False. The correct statement is as follows.**