Verified Answer :

  • Identify the coefficient of the linear term as \(B=1\) and the constant term as \(C=-20\).
  • Find two numbers that add up to \(1\) and multiply to \(-20\), which are \(5\) and \(-4\).
  • Express the quadratic trinomial in the factored form \((x + m)(x + n)\).
  • The correct factored expression is \(\boxed{(x + 5)(x - 4)}\).

Explanation

  1. Analysis of the Quadratic Expression

To factor the quadratic expression \(x^2 + x - 20\), we look for two numbers, \(m\) and \(n\), such that their sum equals the coefficient of the linear term (\(x\)) and their product equals the constant term. The quadratic expression is in the form \(x^2 + Bx + C\), where \(B = 1\) and \(C = -20\). Therefore, we need to find \(m\) and \(n\) such that:
\[m + n = 1\]
\[mn = -20\]

  1. Finding the Correct Factors

We need to identify two numbers whose product is \(-20\) and whose sum is \(1\). Let's list the factor pairs of \(-20\):

  • \(1\) and \(-20\): Sum is \(1 + (-20) = -19\)
  • \(-1\) and \(20\): Sum is \(-1 + 20 = 19\)
  • \(2\) and \(-10\): Sum is \(2 + (-10) = -8\)
  • \(-2\) and \(10\): Sum is \(-2 + 10 = 8\)
  • \(4\) and \(-5\): Sum is \(4 + (-5) = -1\)
  • \(-4\) and \(5\): Sum is \(-4 + 5 = 1\)

The pair \(-4\) and \(5\) satisfies both conditions: \(-4 \times 5 = -20\) and \(-4 + 5 = 1\).

  1. Conclusion and Final Answer

Using the values \(m = 5\) and \(n = -4\), we can write the quadratic expression as:
\[x^2 + x - 20 = (x + 5)(x - 4)\]
Comparing this result with the provided options, we see that it matches the third option \((x + 5)(x - 4)\). Thus, the correct factors are \((x + 5)(x - 4)\). \(\boxed{(x + 5)(x - 4)}\).

Examples

Factoring quadratic expressions is a fundamental skill in algebra with many real-life applications. For example, if you are designing a rectangular garden and know that the area can be represented by the expression \(x^2 + x - 20\) square meters, you can factor this expression to find the dimensions of the garden. By setting the area to \((x+5)(x-4)\), you can easily determine that the length and width of the garden are \(x+5\) and \(x-4\) meters respectively, which helps in planning and purchasing materials like fencing.