## Simplifying Complex Algebraic Expressions

This article will guide you through the process of simplifying the complex algebraic expression:

**(x^2/x^3-4x+6/6-3x+1/x+2) (x-2+10-x^2/x+2)**

### Step 1: Simplifying Individual Fractions

First, we need to simplify each individual fraction within the expression:

**x^2/x^3:**This simplifies to**1/x**.**6/6:**This simplifies to**1**.**1/x+2:**This remains as**1/(x+2)**.**10-x^2/x+2:**This remains as**(10-x^2)/(x+2)**.

Now, the expression becomes:

**(1/x-4x+1-3x+1/(x+2)) (x-2+(10-x^2)/(x+2))**

### Step 2: Combining Like Terms

Next, we combine like terms within the parentheses:

**(1/x - 4x - 3x + 1) (x - 2 + (10 - x^2)/(x + 2))****(1/x - 7x + 1) (x - 2 + (10 - x^2)/(x + 2))**

### Step 3: Finding a Common Denominator

To further simplify, we need to find a common denominator for the fractions within each set of parentheses:

**(1/x - 7x^2/x + x/x) (x - 2 + (10 - x^2)/(x + 2))****((1 - 7x^2 + x)/x) ((x(x + 2))/(x + 2) - 2(x + 2)/(x + 2) + (10 - x^2)/(x + 2))**

### Step 4: Combining Numerators

Now, we can combine the numerators over the common denominators:

**((1 - 7x^2 + x)/x) ((x^2 + 2x - 2x - 4 + 10 - x^2)/(x + 2))****((1 - 7x^2 + x)/x) (6/(x + 2))**

### Step 5: Multiplying the Fractions

Finally, we multiply the two fractions together:

**(1 - 7x^2 + x)(6) / (x(x + 2))****(6 - 42x^2 + 6x) / (x^2 + 2x)**

### Final Simplified Expression

The simplified form of the original expression is:

**(6 - 42x^2 + 6x) / (x^2 + 2x)**

**Important Note:** This expression can be further simplified by factoring out a 6 from the numerator, but it is not necessary for this solution.