## Simplifying the Expression: (x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4)

This expression may seem daunting at first, but we can simplify it using a few algebraic techniques. Here's how:

### Recognizing Patterns

The expressions within the parentheses are special:

**(1+x-x^2+x^3-x^4):**This is a finite geometric series with the first term being 1 and the common ratio being -x.**(1+x+x^2+x^3+x^4):**This is also a finite geometric series, with the first term being 1 and the common ratio being x.

### Applying the Geometric Series Formula

The sum of a finite geometric series is given by:

**S = a(1-r^n)/(1-r)**

where:

- S is the sum of the series
- a is the first term
- r is the common ratio
- n is the number of terms

Let's apply this to our expression:

**For (1+x-x^2+x^3-x^4):**a = 1, r = -x, n = 5. Therefore: S = 1(1-(-x)^5)/(1-(-x)) = (1+x^5)/(1+x)**For (1+x+x^2+x^3+x^4):**a = 1, r = x, n = 5. Therefore: S = 1(1-x^5)/(1-x)

### Substituting and Simplifying

Now, let's substitute these simplified expressions back into the original expression:

(x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4) = (x+2)((1+x^5)/(1+x)) - (1-x)((1-x^5)/(1-x))

Notice that the denominators (1+x) and (1-x) cancel out:

= (x+2)(1+x^5) - (1-x)(1-x^5)

Now, we can expand the expressions:

= x + 2 + x^6 + 2x^5 - (1 - x - x^5 + x^6)

Finally, combining like terms:

= **3x + 3x^5 + 1**

### Conclusion

The simplified form of the expression (x+2)(1+x-x^2+x^3-x^4)-(1-x)(1+x+x^2+x^3+x^4) is **3x + 3x^5 + 1**. By recognizing the patterns of geometric series and applying the formula, we were able to simplify the expression considerably.