(x+2)^2+4(x+2)+3

3 min read Jun 16, 2024

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.