## Solving the Equation: ((x+2)(x^(2)-2x+1))/(4+3x-x^(2)) = 0

This equation presents a rational function set equal to zero. To solve for *x*, we can utilize the following steps:

### Understanding the Equation

**Numerator:**The numerator is factored into (x+2) and (x^(2)-2x+1). The second factor is a perfect square trinomial: (x-1)^(2)**Denominator:**The denominator is a quadratic expression. We can factor it as well.

### Solving for x

**Set the numerator equal to zero:**Since a fraction equals zero only when the numerator is zero, we have: (x+2)(x-1)^(2) = 0**Solve for x:**This equation has two solutions:- x + 2 = 0 =>
**x = -2** - (x-1)^(2) = 0 =>
**x = 1**

- x + 2 = 0 =>

### Verifying the Solutions

It's crucial to check if these solutions make the denominator equal to zero. If they do, they are **extraneous solutions** and must be discarded.

**For x = -2:**4 + 3(-2) - (-2)^(2) = 4 - 6 - 4 = -6 ≠ 0. Therefore,**x = -2 is a valid solution**.**For x = 1:**4 + 3(1) - (1)^(2) = 4 + 3 - 1 = 6 ≠ 0. Therefore,**x = 1 is a valid solution**.

### Conclusion

The solutions to the equation ((x+2)(x^(2)-2x+1))/(4+3x-x^(2)) = 0 are **x = -2** and **x = 1**. These solutions have been verified and are not extraneous.