## Expanding (a+b+c+d+e+f+g+h+i)^2

The expression (a+b+c+d+e+f+g+h+i)^2 represents the square of the sum of nine variables. Expanding this expression can be quite tedious, but there's a systematic way to do it.

### Understanding the Concept

The expression (a+b+c+d+e+f+g+h+i)^2 is essentially multiplying the sum of the variables by itself:

(a+b+c+d+e+f+g+h+i) * (a+b+c+d+e+f+g+h+i)

To expand this, we need to distribute each term in the first set of parentheses to every term in the second set.

### Using the FOIL Method (Generalized)

While the FOIL method is commonly used for expanding binomials, we can generalize it to handle multiple variables. The process involves:

**First:**Multiply the first terms of each set of parentheses.**Outer:**Multiply the outer terms of each set of parentheses.**Inner:**Multiply the inner terms of each set of parentheses.**Last:**Multiply the last terms of each set of parentheses.

**However**, with nine variables, this will result in a lot of terms!

### The Resulting Expression

After expanding the entire expression, we get a sum of squares and cross-product terms:

**(a+b+c+d+e+f+g+h+i)^2 =**

**a²**+**b²**+**c²**+**d²**+**e²**+**f²**+**g²**+**h²**+**i²****+ 2ab + 2ac + 2ad + 2ae + 2af + 2ag + 2ah + 2ai****+ 2bc + 2bd + 2be + 2bf + 2bg + 2bh + 2bi****+ 2cd + 2ce + 2cf + 2cg + 2ch + 2ci****+ 2de + 2df + 2dg + 2dh + 2di****+ 2ef + 2eg + 2eh + 2ei****+ 2fg + 2fh + 2fi****+ 2gh + 2gi****+ 2hi**

This expanded expression consists of:

**Nine squared terms:**Each variable multiplied by itself.**36 cross-product terms:**Each unique combination of two distinct variables, multiplied by 2.

### Key Points:

- The number of terms in the expanded expression grows rapidly with the number of variables.
- This expansion can be used to simplify expressions involving squared sums.
- It's crucial to be organized and methodical when expanding such expressions to avoid errors.