To prove that the two expressions are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.
Expression 1: (x - 5)(x + 3)
Using the Distributive Property (also known as the FOIL method for binomials), we can expand:
\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]
So, Expression 1 simplifies to:
Expression 1 Result: \(x^2 - 2x - 15\)
Expression 2: 2x^2 - x^2 - 2x - 20 + 5
Now, simplify Expression 2:
\[ 2x^2 - x^2 - 2x - 20 + 5 \] \[ = (2x^2 - x^2) - 2x + (5 - 20) \] \[ = x^2 - 2x - 15 \]
So, Expression 2 simplifies to:
Expression 2 Result: \(x^2 - 2x - 15\)
Conclusion
Both expressions simplify to \(x^2 - 2x - 15\), thus proving that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent.