To find expressions that simplify to the trinomial \( 4x^2 - 20x - 24 \), we can start by factoring the trinomial. First, we can factor out the greatest common factor (GCF) from all terms.
- Identify the GCF of the coefficients: The coefficients are 4, -20, and -24. The GCF is 4.
- Factor out the GCF from the trinomial:
\[ 4x^2 - 20x - 24 = 4(x^2 - 5x - 6) \]
Next, we need to factor the quadratic trinomial \( x^2 - 5x - 6 \):
To factor \( x^2 - 5x - 6 \), we look for two numbers that multiply to \(-6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-6\) and \(1\) satisfy these requirements because:
\[ -6 \times 1 = -6 \quad \text{and} \quad -6 + 1 = -5 \]
Thus, we can factor \( x^2 - 5x - 6 \) as:
\[ x^2 - 5x - 6 = (x - 6)(x + 1) \]
Now we can substitute back into our expression:
\[ 4(x^2 - 5x - 6) = 4(x - 6)(x + 1) \]
So one expression that produces \( 4x^2 - 20x - 24 \) when simplified is:
\[ 4(x - 6)(x + 1) \]
In summary, a simplified expression that produces \( 4x^2 - 20x - 24 \) is \( 4(x - 6)(x + 1) \). There are infinitely many equivalent forms (such as rearranging terms or multiplying out), but this is a straightforward one.
1 answer