Verified Answer :

  • Use the identity \((a+b)^2 = a^2 + 2ab + b^2\) for the expression \((2x+3)^2\).
  • Substitute \(a=2x\) and \(b=3\) to get \((2x)^2 + 2(2x)(3) + 3^2\).
  • Simplify each term to arrive at \(4x^2 + 12x + 9\).
  • Match the result with the given options, identifying the final answer as \(\boxed{4x^2+12x+9}\).

Explanation

  1. Analysis of the problem

The given expression is \((2x+3)^2\). This is in the form of the square of a binomial, \((a+b)^2\), where \(a = 2x\) and \(b = 3\). The identity for expanding such an expression is \((a+b)^2 = a^2 + 2ab + b^2\). Our goal is to perform this expansion and match it with the provided options.

  1. Expansion and calculation

We apply the identity \((a+b)^2 = a^2 + 2ab + b^2\) with \(a = 2x\) and \(b = 3\):

  1. Compute the first term \(a^2 = (2x)^2 = 4x^2\).
  2. Compute the middle term \(2ab = 2(2x)(3) = 2 \cdot 6x = 12x\).
  3. Compute the last term \(b^2 = 3^2 = 9\).

Combining these parts, we get \(4x^2 + 12x + 9\).

  1. Conclusion

Comparing our result, \(4x^2 + 12x + 9\), with the provided options:

  • \(4x^2+12x+9\)
  • \(4x+6\)
  • \(4x^2+6x+9\)
  • \(4x^2+9\)

The calculated result corresponds to the first option. Therefore, the expanded form is \(\boxed{4x^2+12x+9}\).

Examples

Expanding binomials is a fundamental skill in algebra, used frequently in physics and engineering. For instance, if you are designing a square parking area and want to increase each side by 3 meters, you would calculate the new area using \((x+3)^2 = x^2 + 6x + 9\). This allows you to quickly determine how much extra material, like concrete or paving stones, you will need for the expansion.

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