## Solving the Equation (x+5)^4 - 10(x+5)^2 + 9 = 0

This equation may look complex at first glance, but it can be solved using a clever substitution.

### Understanding the Structure

The equation features a pattern: both terms with 'x' are raised to even powers. This suggests a substitution to simplify the equation.

### Substitution Technique

Let's introduce a new variable: **y = (x+5)^2**.

Now, we can rewrite the equation in terms of 'y':

**y^2 - 10y + 9 = 0**

This is a simple quadratic equation, which we can solve using various methods, like factoring or the quadratic formula.

### Solving the Quadratic Equation

In this case, the quadratic equation factors easily:

**(y - 9)(y - 1) = 0**

This gives us two solutions for 'y':

**y = 9****y = 1**

### Back to 'x'

Now, we need to substitute back to find the values of 'x'. Remember, **y = (x+5)^2**.

**Case 1: y = 9**

(x+5)^2 = 9

Taking the square root of both sides:

x + 5 = ±3

Solving for 'x':

**x = -2****x = -8**

**Case 2: y = 1**

(x+5)^2 = 1

Taking the square root of both sides:

x + 5 = ±1

Solving for 'x':

**x = -4****x = -6**

### Conclusion

Therefore, the solutions to the equation (x+5)^4 - 10(x+5)^2 + 9 = 0 are:

**x = -2****x = -8****x = -4****x = -6**