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When we talk about binomial expansion, we're discussing a way to multiply out expressions that are raised to a power. The term 'binomial' means that the expression includes two terms. For example, in our exercise, the binomial is \((x+1)\).
When we expand a binomial, we use specific algebraic rules to distribute the terms.
The most common formula used for binomial expansion is the square of a binomial formula: \((a+b)^2 = a^2 + 2ab + b^2\).
This formula helps us take the guesswork out of multiplying the binomial out step by step. In our given exercise, \((x+1)^2\), we already know that \a = x\ and \b = 1\. Using the binomial expansion formula, we can make this process straightforward.

Algebraic formulas are pre-set rules that help us perform complex calculations swiftly. One important algebraic formula relevant to this exercise is the square of a binomial formula.
Let's dive deeper into this formula: \((a+b)^2 = a^2 + 2ab + b^2\).
The formula tells us how to expand and simplify a binomial expression quickly.

• In the formula, \a^2\ means we square the first term.
• \2ab\ means we double the product of the two terms.
• \b^2\ means we square the second term.

By using these rules, you save time and make fewer mistakes compared to multiplying out the binomial by hand. In our example, we used \a=x\ and \b=1\, and applied the formula to get \x^2 + 2(x)(1) + 1^2\.

After expanding a binomial, our next task is to simplify the expression. Simplifying means combining like terms and making the expression as concise as possible.
Let's review the simplification process from our exercise: \(x+1)^2 = x^2 + 2(x)(1) + 1^2\).
In the expanded form, we see that we have \x^2\, \2x\, and \1\.
These are all like terms, meaning they can be combined for simplicity.
When we put them together, we get \x^2 + 2x + 1\, which is the simplified form of the original binomial squared.
Remember that simplifying expressions makes them easier to work with in further algebra problems.