Method One:

We will start expanding slowly, and we will end up at exponent 

Step 1: Expand: 





Step 2: Multiply  by the product of . By doing this, we are now expanding .





Step 3: Multiply by  again




.

After Step 4: 
After Step 5: 

After Step 6, the final answer is:

.

Method Two:

You can find the expansion of this binomial by using the Pascal's Triangle (shown below)

If you look at Row  of the triangle above, the row that starts with .

We need to negate every nd term, as the answer in Method One has every even term negative. 

We will still get the answer: .

Step 1: Expand 



Step 2: FOIL the first two parentheses:



Step 3: Multiply the expansion in step 2 by :



The expanded form of  is .

Step 1: Multiply 



Step 2: Multiply the result in step 1 by 



Step 3: Multiply the result of step 2 by 


Simplify:

The easiest way to expand binomials raised to higher powers is to use Pascal's Triangle. 

Pascal's Triangle is used to find the multipliers for each level of exponent. 

It follows the pattern listed below. 

To complete the expansion we will take the row that corresponds to the 4th exponent for this problem. 

We will now organize this into columns and rows.

The second and third rows are organized by taking the left term from the highest to lowest power, and the right term from lowest to highest power. 

Since the x is on the left, it is raised to the 4th power, 3rd power and so on. 

Since the 3 is on the right, it is raised to the 0 power, 1 power and so on. 

We now simplify each of the terms, the bottom row is the only one to be simplified in this case. Anything to the zero power except zero is 1. 

Now we multiply each column together to obtain the full expansion. 

For example to obtain the third term we multiplied everything in the 3rd column: 

We did this for all of the columns to get the below final answer. 

Let's start with a smaller expansion:

We multiply the expansion of  by :

Multiply again by :

Multiply by :

Expand by distributing each of the factors

Simplify

Simplify

Step 1: Let's start small, expand .

Step 2: Expand 

Take the final answer of Step 1 and multiply it by ...

Step 3: Multiply again by  to the final answer of Step 2...

The corollary to the Fundamental Theorem of Algebra states that for any polynomial the number of solutions will match the degree of the function.

The degree of a function is determined by the highest exponent for x, which in this case is 7.

This means that there will be 7 solutions total for the below function.

This means that max number of REAL solutions would be 7, but the total number of solutions, real, repeated or irrational will total 7. 

In order to determine the correct answer, we must first change the function to be in standard form. A function in standard form begins with the largest exponent then decreases from there. 

We must change:

 to become: 

Once we have established standard form, we can now see that this is a degree 3 polynomial, which means that it will have 3 roots or solutions.