Verified Answer :

  • Use the binomial expansion identity: \((a + b)^2 = a^2 + 2ab + b^2\).
  • Substitute \(a = 2x\) and \(b = 3\) into the formula to get \((2x)^2 + 2(2x)(3) + 3^2\).
  • Simplify each term to arrive at \(4x^2 + 12x + 9\).
  • The final result is \(\boxed{4x^2 + 12x + 9}\).

Explanation

  1. Understanding the Expansion Identity

We start by identifying the expression as the square of a binomial: \((2x + 3)^2\). To expand this expression, we use the standard algebraic identity for the square of a sum:
\[(a + b)^2 = a^2 + 2ab + b^2\]
In our specific case, let \(a = 2x\) and \(b = 3\). By substituting these values into the identity, we can systematically expand the binomial.

  1. Calculating the Expanded Terms

Now we calculate each part of the identity separately using \(a = 2x\) and \(b = 3\):

  1. The first term is \(a^2 = (2x)^2 = 4x^2\).
  2. The middle term is \(2ab = 2 \times (2x) \times 3 = 12x\).
  3. The last term is \(b^2 = 3^2 = 9\).

Combining these parts, we get the expanded form of the original expression.

  1. Finalizing the Expression

By adding the calculated components together, we obtain:
\[(2x + 3)^2 = 4x^2 + 12x + 9\]
This matches the desired form \(Ax^2 + Bx + C\), where \(A = 4\), \(B = 12\), and \(C = 9\). Thus, the expansion is \(\boxed{4}x^2 + \boxed{12}x + \boxed{9}\).

Examples

Imagine you are designing a square garden bed that needs to be expanded. If the original side length is \(x\) meters, and you increase the length of each side by a fixed amount to make it \((2x + 3)\) meters, the area of the new square garden will be \((2x + 3)^2\) square meters. Expanding this expression to \(4x^2 + 12x + 9\) helps you calculate the total area of the garden based on different values of \(x\), allowing you to determine exactly how much soil, fertilizer, or fencing you need for the project.

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