\( \frac { x + 2 } { x ^ { 2 } - 3 x - 4 } = \frac { 3 } { x - 4 } \quad - \frac { 1 } { 2 x + 2 } \)

To solve the equation, start by simplifying both sides. The left side can be written with a common denominator, which is \( (x^2 - 3x - 4)(2x + 2) \). The right side also needs to be simplified to have a common denominator. Next, cross-multiply to eliminate the fractions. You'll get a polynomial equation that can be simplified and solved for \( x \). Remember to check for extraneous solutions by substituting back into the original equation, as some solutions may create undefined terms. After simplifying and solving, you may arrive at specific values for \( x \). Make sure to examine your solutions to confirm they don’t make any denominators zero, as that would invalidate the solution! Happy solving!