Rational equation solving (SAT). Game on.

For $x \neq -7$, the factor $x+7$ cancels, giving $x+1 = -2$, so $x = -3\textsf{.}$

For $x \neq 3$, the factor $x-3$ cancels, giving $x+5 = 12$, so $x = 7\textsf{.}$

For $x \neq -2$, the factor $x+2$ cancels, giving $x-8 = 5$, so $x = 13\textsf{.}$

Factor the numerator: $x^{2} - 25 = (x-5)(x+5)$. For $x \neq -5$, the factor $x+5$ cancels, giving $x-5 = 2$, so $x = 7\textsf{.}$

Factor the numerator: $x^{2} - 16 = (x-4)(x+4)$. For $x \neq 4$, the factor $x-4$ cancels, giving $x+4 = 9$, so $x = 5\textsf{.}$

Factor the numerator: $x^{2} - 49 = (x-7)(x+7)$. For $x \neq 7$, the factor $x-7$ cancels, giving $x+7 = 6$, so $x = -1\textsf{.}$

Factor the numerator as a perfect square: $x^{2} + 6x + 9 = (x+3)^{2}$. For $x \neq -3$, one factor of $x+3$ cancels, giving $x+3 = 4$, so $x = 1\textsf{.}$

Factor the numerator as a perfect square: $x^{2} + 10x + 25 = (x+5)^{2}$. For $x \neq -5$, one factor of $x+5$ cancels, giving $x+5 = 3$, so $x = -2\textsf{.}$