To prove that the expressions \( (x−5)(x+3) \) and \( 2x^2−x^2−2x−20+5 \) are equivalent using the Distributive Property, we can expand Expression 1 and simplify Expression 2.
Expression 1:
Using the Distributive Property (also known as FOIL for binomials), we expand \( (x−5)(x+3) \):
\[ (x−5)(x+3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \]
So, \( (x−5)(x+3) \) simplifies to \( x^2 - 2x - 15 \).
Expression 2:
Now we simplify \( 2x^2−x^2−2x−20+5 \):
\[ 2x^2 - x^2 - 2x - 20 + 5 = (2x^2 - x^2) - 2x + (5 - 20) = x^2 - 2x - 15 \]
Thus, \( 2x^2−x^2−2x−20+5 \) simplifies to \( x^2 - 2x - 15 \).
Conclusion:
Since both expressions simplify to \( x^2 - 2x - 15 \), we have shown that \( (x−5)(x+3) \) is equivalent to \( 2x^2−x^2−2x−20+5 \).