To prove that the expressions \( (x - 5)(x + 3) \) and \( 2x^2 - x^2 - 2x - 20 + 5 \) are equivalent using the Distributive Property, we will expand Expression 1 and simplify Expression 2.
Step 1: Expand Expression 1
Using the Distributive Property (FOIL method):
\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \]
Calculating each part, we have:
\[ = x^2 + 3x - 5x - 15 \]
Combining like terms:
\[ = x^2 - 2x - 15 \]
Step 2: Simplify Expression 2
Now we simplify Expression 2:
\[ 2x^2 - x^2 - 2x - 20 + 5 \]
Calculating \((2x^2 - x^2)\):
\[ = (2 - 1)x^2 - 2x - 20 + 5 \]
This simplifies to:
\[ = x^2 - 2x - 20 + 5 \]
Combining the constant terms \((-20 + 5)\):
\[ = x^2 - 2x - 15 \]
Conclusion
Both expressions simplify to \( x^2 - 2x - 15 \).
Thus, we can conclude:
Expression 1 simplifies to \( x^2 - 2x - 15 \), and Expression 2 also simplifies to \( x^2 - 2x - 15 \), proving that both expressions are equivalent.