Verified Answer :

  • Identify the expression as a squared binomial \((x+1)^2\).
  • Apply the identity \((a+b)^2 = a^2 + 2ab + b^2\) with \(a=x\) and \(b=1\).
  • Calculate the expansion: \(x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1\).
  • The final expansion is \(\boxed{x^2 + 2x + 1}\).

Explanation

  1. Analysis of the expression to expand

The problem asks to expand the expression \((x+1)^2\). This is a square of a binomial expression. To solve this, we can use the general algebraic identity for squaring a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). This formula allows us to square any binomial by calculating the square of the first term, twice the product of the two terms, and the square of the second term.

  1. Applying the algebraic identity

By comparing \((x+1)^2\) to the identity \((a+b)^2\), we can identify \(a = x\) and \(b = 1\). Applying the formula, we substitute these values:

  • The first term squared: \(a^2 = x^2\)
  • Twice the product of the terms: \(2ab = 2 \times x \times 1 = 2x\)
  • The second term squared: \(b^2 = 1^2 = 1\)
  1. Combining the results

Now, we add these three components together to find the expanded form:
\[x^2 + 2x + 1\]
This matches the third option provided: \(x^2+2x+1\). Thus, the correct expansion of the expression \((x+1)^2\) is \(x^2 + 2x + 1\).

Examples

In many real-life situations, we encounter the need to calculate the area of a square whose side length is an expression. For instance, if you are designing a garden square with a side length of \((x+1)\) meters, the total area of the garden is given by \((x+1)^2\). Using the expansion \((x^2 + 2x + 1)\), you can easily determine how the total area changes as the base unit \(x\) changes, helping in planning and resource allocation.

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