Verified Answer :

  • Use the distributive property (FOIL) to expand the product of the two binomials: \((2x)(4x) + (2x)(3) + (-5)(4x) + (-5)(3)\).
  • Calculate each product to get \(8x^2 + 6x - 20x - 15\).
  • Combine the like terms \(6x\) and \(-20x\) to simplify the expression.
  • The final simplified expression is \(\boxed{8x^2 - 14x - 15}\).

Explanation

  1. Problem Analysis

The objective is to expand the product of two binomials, \((2x - 5)(4x + 3)\). This expression requires the use of the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). Our goal is to represent this as a single quadratic expression in standard form, \(ax^2 + bx + c\).

  1. Applying the Distributive Property (FOIL)

To multiply the two binomials, we distribute each term in the first binomial to each term in the second binomial:

  1. First: Multiply the first terms of each binomial: \(2x \times 4x = 8x^2\).
  2. Outer: Multiply the outer terms: \(2x \times 3 = 6x\).
  3. Inner: Multiply the inner terms: \(-5 \times 4x = -20x\).
  4. Last: Multiply the last terms: \(-5 \times 3 = -15\).

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

  1. Combining Like Terms and Simplifying

Now, we combine the like terms, which are the terms containing \(x\). We have \(6x\) and \(-20x\):
\[6x - 20x = -14x\]
Substituting this back into our expression, we arrive at the final simplified quadratic polynomial:
\[8x^2 - 14x - 15\]

Examples

Expanding binomials is a foundational skill used in many practical applications. For instance, if you are designing a rectangular garden where the length is represented by \((2x - 5)\) meters and the width by \((4x + 3)\) meters, the area of the garden is given by the product \((2x - 5)(4x + 3)\) square meters. Calculating this expansion helps you determine the total area as a function of \(x\), which is useful for tasks like calculating the amount of soil or fertilizer needed based on the garden's dimensions.

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