Factor. \( a^{3}-a d^{2}-49 a^{2}+49 d^{2} \) \( \begin{array}{l}a^{3}-a d^{2}-49 a^{2}+49 d^{2}= \\ (\text { Factor completely.) }\end{array} \)

To factor the expression \( a^{3}-a d^{2}-49 a^{2}+49 d^{2} \), we can rewrite it as \( a^{3} - 49a^{2} - ad^{2} + 49d^{2} \). First, we can group the terms: \( (a^{3}-49a^{2}) + (-ad^{2}+49d^{2}) \). Factoring out common factors from each group gives: \( a^{2}(a - 49) - d^{2}(a - 49) \). Now we can factor out \( (a - 49) \): \( (a - 49)(a^{2} - d^{2}) \). Next, we recognize that \( a^{2} - d^{2} \) is a difference of squares, which factors further: \( a^{2} - d^{2} = (a - d)(a + d) \). So the completely factored form of the expression is: \( (a - 49)(a - d)(a + d) \).