## Solving the Quadratic Equation: (x^2 + x)^2 - 8(x^2 + x) + 12 = 0

This equation might look intimidating at first glance, but it can be solved using a simple substitution and factoring.

### The Substitution Trick

Notice that the expression **(x² + x)** appears multiple times in the equation. We can simplify the problem by substituting a new variable, say **y**, for this expression:

Let **y = x² + x**

Now, our equation becomes:

**y² - 8y + 12 = 0**

### Factoring the Quadratic Equation

This is a standard quadratic equation, which we can solve by factoring. We need to find two numbers that add up to -8 and multiply to 12. These numbers are -6 and -2:

**y² - 6y - 2y + 12 = 0**

**y(y - 6) - 2(y - 6) = 0**

**(y - 6)(y - 2) = 0**

Therefore, **y = 6** or **y = 2**.

### Substituting Back and Solving for x

Now we need to substitute back the original expression for y:

**Case 1: y = 6**

**x² + x = 6**

**x² + x - 6 = 0**

Factoring this quadratic equation, we get:

**(x + 3)(x - 2) = 0**

Therefore, **x = -3** or **x = 2**.

**Case 2: y = 2**

**x² + x = 2**

**x² + x - 2 = 0**

Factoring this quadratic equation, we get:

**(x + 2)(x - 1) = 0**

Therefore, **x = -2** or **x = 1**.

### Final Solutions

The solutions to the equation (x² + x)² - 8(x² + x) + 12 = 0 are:

**x = -3****x = 2****x = -2****x = 1**