### Instruction: Find all solutions to the equation \[\frac{\left(\frac{x}{x+1}\right)^2 + 11}{\left(\frac{x}{x+1}\right)^2 + 1} = 2.\]Enter all the solutions, separated by commas.### Response: This is a quadratic equation in disguise, because the expression \[\frac{x}{x+1}\] is a linear function of x that I can replace with a new variable, say y.
Then the equation becomes \[\frac{y^2 + 11}{y^2 + 1} = 2.\]
I can cross-multiply to get rid of the fractions and simplify the equation.
I get \[y^2 + 11 = 2y^2 + 2.\]
Subtracting y^2 and 2 from both sides, I get \[y^2 - 9 = 0.\]
This is a difference of squares, so I can factor it as \[(y - 3)(y + 3) = 0.\]
This means that y can be either 3 or -3.
But I want to find the solutions for x, not y.
To do that, I can plug in y = 3 and y = -3 into the original expression and solve for x. I get \[\frac{x}{x+1} = 3 \implies x = 3x + 3 \implies -2x = 3 \implies x = -\frac{3}{2}\] and \[\frac{x}{x+1} = -3 \implies x = -3x - 3 \implies 4x = -3 \implies x = -\frac{3}{4}.\]
So the solutions for x are -3/2 and -3/4.