Solve. Express answer using interval notation where appropriate. Check for extraneous solutions.
\(\frac{3x}{x+1}+\frac{5}{x-2}=\frac{15}{x^2-x-2}\)
Express \(x^2-x-2 \text{ as }(x+1)(x-2)\)
Multiply every term by \((x+1)(x-2) \text{ to get } 3x(x-2)+5(x+1)=15\)
Collect terms: \(3x^2-x-10=0\)
Solve the quadratic to get \(x=-\frac{5}{3}\text{ and }x=2\)
2 is an extraneous solution (try substituting 2 into the original equation and see what you get!).