## Factoring and Solving (x+2)^2 + 4(x+2) + 3

This expression, (x+2)^2 + 4(x+2) + 3, appears complex, but can be easily factored and solved by understanding a few key concepts:

### Recognizing the Pattern

The expression is structured in a way that resembles a quadratic equation. Notice the repeated term "(x+2)". We can simplify this by making a substitution.

**Let:** u = (x+2)

**Now our expression becomes:** u^2 + 4u + 3

### Factoring the Quadratic

The quadratic expression u^2 + 4u + 3 can be factored by finding two numbers that add up to 4 (the coefficient of the middle term) and multiply to 3 (the constant term).

These numbers are 1 and 3:

- 1 + 3 = 4
- 1 * 3 = 3

Therefore, we can factor the quadratic as: (u + 1)(u + 3)

### Back-Substituting

Now, let's substitute back our original value for u:

**(u + 1)(u + 3) becomes ((x+2) + 1)((x+2) + 3)**

### Simplifying

Finally, we can simplify the expression:

**(x + 3)(x + 5)**

### Finding the Solutions

To find the solutions (also called roots or zeros), we set the expression equal to zero and solve for x:

- (x + 3)(x + 5) = 0

This means either (x + 3) = 0 or (x + 5) = 0.

**Therefore, x = -3 or x = -5**

### Conclusion

We've successfully factored and solved the expression (x+2)^2 + 4(x+2) + 3 by using substitution, recognizing the quadratic pattern, and applying basic factorization techniques. The solutions to this expression are **x = -3** and **x = -5**.