## Factoring and Exploring (x+3)(x+4)(x+5)(x+6)+1

The expression (x+3)(x+4)(x+5)(x+6)+1 might seem complex at first glance, but with some clever manipulation, we can uncover its hidden beauty and explore its fascinating properties.

### Factoring by Grouping

The key to factoring this expression lies in grouping the terms strategically. Let's rearrange the terms and observe:

(x+3)(x+6) * (x+4)(x+5) + 1

Notice that the first two factors and the last two factors share a common pattern. This allows us to apply the difference of squares factorization:

[(x+3)(x+6) + 1][(x+4)(x+5) + 1] - 1

Now, we expand the products inside the brackets:

[(x² + 9x + 18) + 1][(x² + 9x + 20) + 1] - 1

This simplifies to:

(x² + 9x + 19)(x² + 9x + 21) - 1

### Final Factored Form

Finally, we can apply the difference of squares factorization one more time to obtain the fully factored form:

**[(x² + 9x + 20) - 1][(x² + 9x + 20) + 1]**

**This simplifies to:**

**(x² + 9x + 19)(x² + 9x + 21)**

### Exploring the Properties

We have successfully factored the expression. This factored form reveals some interesting properties:

**Symmetry:**Notice the symmetry in the factors. Both factors are quadratic expressions with the same linear term (9x). This symmetry suggests potential connections to other mathematical concepts.**Roots:**Finding the roots of the expression involves setting each factor equal to zero and solving the resulting quadratic equations. This will give us four solutions for the original expression.**Graphing:**The factored form can be used to graph the function represented by the expression. The roots will correspond to the x-intercepts of the graph, and the symmetry will be evident in the shape of the curve.

### Conclusion

By strategically factoring and manipulating the expression (x+3)(x+4)(x+5)(x+6)+1, we have uncovered its fascinating structure and gained insights into its properties. This journey showcases the power of algebraic manipulation and the beauty of mathematical patterns.