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The difference of squares is a powerful algebraic concept that frequently appears when multiplying certain binomials. This property states that for any two numbers or expressions, if you have

• a - b
• a + b

Their product can be simplified to the form \( a^2 - b^2 \).
This is because the mixed terms cancel each other out naturally, essentially reducing the work you have to do. In the given exercise, the expression \((x-5)(x+5)\) is a classic instance of the difference of squares. Here, if you let \( a = x \) and \( b = 5 \), then applying the formula gives us \( x^2 - 25 \). This transforms the initial expression into something much simpler to manage.

The distributive property is a cornerstone of algebra that allows us to multiply a single term across terms inside a set of parentheses effectively. In simpler terms, if you have a term multiplying a sum (or difference) of terms, you can distribute the multiplication over each term inside, simplifying your calculations. For example:

• \( a(b + c) = ab + ac \)

In our exercise, after the application of the difference of squares, we end up with the expression:\((x^2 - 25)(x^2 + 25)\).
Using the distributive property or the FOIL method with these expressions, we first multiply each term in \((x^2 - 25)\) by each term in \((x^2 + 25)\).
This results in \[x^4 + 25x^2 - 25x^2 - 625\]. As you can see, the x-squared terms cancel each other out due to their opposing signs, simplifying the expression further.

Simplifying expressions is the process of consolidating terms to make an algebraic expression easier to interpret and work with. After applying the difference of squares and distributive property in our exercise, we arrived at:\[x^4 + 25x^2 - 25x^2 - 625\].
The like terms, which in this case are \( 25x^2 \) and \( -25x^2 \), cancel each other out, making our job easier. When simplifying, always remember:

• Combine like terms, which are terms that have the same variables raised to the same power.
• Eliminate any zero-sum pairs, terms that add up to zero.

This leaves us with the simplified expression \( x^4 - 625 \). This process not only makes the expression simpler but is crucial in solving polynomial equations or performing additional algebraic operations effectively.