Solving the Equation: (x/(x+1))^2  5(x/(x+1)) + 6 = 0
This equation may look intimidating at first, but it can be solved using a simple substitution technique. Let's break down the steps:
1. Substitution:
Let y = x/(x+1). This simplifies the equation to:
y²  5y + 6 = 0
2. Factoring the Quadratic:
This is now a standard quadratic equation. We can factor it as:
(y  2)(y  3) = 0
This gives us two possible solutions for y:
 y = 2
 y = 3
3. BackSubstituting to Find x:
Now, we need to substitute back our original value for y:

For y = 2:
 2 = x/(x+1)
 2x + 2 = x
 x = 2

For y = 3:
 3 = x/(x+1)
 3x + 3 = x
 2x = 3
 x = 3/2
4. Checking for Extraneous Solutions:
It's crucial to check our solutions in the original equation to ensure they are valid. We find that both solutions, x = 2 and x = 3/2, satisfy the original equation.
Conclusion:
Therefore, the solutions to the equation (x/(x+1))^2  5(x/(x+1)) + 6 = 0 are x = 2 and x = 3/2.