Verified Answer :

  • Apply the distributive property (FOIL method) to the expression: \((2x \cdot 3x) + (2x \cdot -4) + (5 \cdot 3x) + (5 \cdot -4)\).
  • Calculate the individual products to get \(6x^2 - 8x + 15x - 20\).
  • Combine the like terms \(-8x\) and \(15x\) to obtain the simplified result \(\boxed{6x^2 + 7x - 20}\).

Explanation

  1. Understanding the Problem

The expression given for expansion is \((2x + 5)(3x - 4)\). This is a product of two linear binomials. To expand this, we employ the distributive property, often referred to as the FOIL method, which stands for First, Outer, Inner, and Last. This method ensures that each term in the first binomial is multiplied by each term in the second binomial.

  1. Applying the FOIL Method

We apply the FOIL method to the binomials \((2x + 5)\) and \((3x - 4)\) as follows:

  • First terms: \(2x \cdot 3x = 6x^2\)
  • Outer terms: \(2x \cdot (-4) = -8x\)
  • Inner terms: \(5 \cdot 3x = 15x\)
  • Last terms: \(5 \cdot (-4) = -20\)

Combining these results, we get the sum: \(6x^2 - 8x + 15x - 20\).

  1. Simplifying the Expression

To simplify the expression, we combine the like terms, which are the terms containing \(x\):
\(-8x + 15x = 7x\)
Now, combining all the components, we arrive at the final simplified quadratic expression: \(6x^2 + 7x - 20\).

Examples

Expanding binomials is a fundamental skill used in various fields, such as calculating the area of shapes with variable dimensions. For instance, if you have a rectangular garden with length \((3x - 4)\) meters and width \((2x + 5)\) meters, the total area is given by the product \((3x - 4)(2x + 5)\). Expanding this expression allows you to model the area as a quadratic function \(6x^2 + 7x - 20\), which can then be used to determine the necessary dimensions for a desired area.