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Order the steps from first to last that would be used to factor 2x^2 - 3x - 20

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To factor the quadratic expression \(2x^2 - 3x - 20\), we can follow these steps in order:

  1. Identify the coefficients: Recognize \(a = 2\), \(b = -3\), and \(c = -20\) from the expression in the form \(ax^2 + bx + c\).

  2. Multiply \(a\) and \(c\): Calculate \(a \cdot c = 2 \cdot (-20) = -40\).

  3. Find two numbers that multiply to \(ac\) and add to \(b\): Look for two numbers that multiply to \(-40\) and add to \(-3\). The numbers are \(5\) and \(-8\) since \(5 \cdot (-8) = -40\) and \(5 + (-8) = -3\).

  4. Rewrite the middle term using the two numbers found: Rewrite \(-3x\) as \(5x - 8x\). Thus, the expression becomes: \[ 2x^2 + 5x - 8x - 20 \]

  5. Group the terms: Group the terms into two pairs: \[ (2x^2 + 5x) + (-8x - 20) \]

  6. Factor out the greatest common factor (GCF) from each group:

    • From the first group \(2x^2 + 5x\), factor out \(x\): \[ x(2x + 5) \]
    • From the second group \(-8x - 20\), factor out \(-4\): \[ -4(2x + 5) \]

  7. Combine the factored terms: Now, combine the factors: \[ x(2x + 5) - 4(2x + 5) = (2x + 5)(x - 4) \]

  8. Final expression: The final factored form of the expression is: \[ (2x + 5)(x - 4) \]

So, the steps in order are:

  1. Identify coefficients.
  2. Multiply \(a\) and \(c\).
  3. Find two numbers that multiply to \(ac\) and add to \(b\).
  4. Rewrite the middle term using the two numbers found.
  5. Group the terms.
  6. Factor out the GCF from each group.
  7. Combine the factored terms.
  8. Write the final factored form.