## Factoring and Simplifying the Expression (x+3)(x+2)(x+1)+(x+2)(x+1)+(x+1)

This article will guide you through the process of factoring and simplifying the expression: **(x+3)(x+2)(x+1)+(x+2)(x+1)+(x+1)**.

### Factoring by Grouping

The key to simplifying this expression lies in recognizing common factors. Notice that **(x+1)** appears in each term. We can factor this out:

**(x+1)[(x+3)(x+2) + (x+2) + 1]**

Now we can focus on simplifying the expression inside the brackets. Let's expand the first term:

**(x+1)[(x^2 + 5x + 6) + (x+2) + 1]**

Combine like terms:

**(x+1)[x^2 + 6x + 9]**

### Final Simplification

The expression inside the brackets is now a perfect square trinomial: **(x+3)^2**

Therefore, the fully factored and simplified expression is:

**(x+1)(x+3)^2**

### Conclusion

By strategically factoring out common terms and recognizing patterns, we've simplified the expression from a seemingly complex form to a concise and factored form: **(x+1)(x+3)^2**. This process demonstrates the power of algebraic manipulation in simplifying expressions and revealing their underlying structure.