To prove that the two algebraic expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent using the Distributive Property, we'll expand Expression 1 and simplify Expression 2.

Question

**Expression 1: \((x - 5)(x + 3)\)**
Using the Distributive Property (also known as the FOIL method for binomials), we distribute each term in the first binomial by each term in the second binomial:

\[
(x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3
\]
\[
= x^2 + 3x - 5x - 15
\]
\[
= x^2 - 2x - 15
\]

**Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\)**
Now, we simplify Expression 2 by combining like terms:

\[
2x^2 - x^2 = (2 - 1)x^2 = x^2
\]
\[
-2x
\]
\[
-20 + 5 = -15
\]
Thus, combining everything, we have:
\[
Expression 2 = x^2 - 2x - 15
\]

Now, since both Expression 1 and Expression 2 simplify to \(x^2 - 2x - 15\), we conclude that the two expressions are equivalent.

**Summary:**
Expression 1 simplifies to \(x^2 - 2x - 15\) via the Distributive Property. Expression 2 simplifies to \(x^2 - 2x - 15\) by combining like terms; therefore, both expressions are equivalent.
make tis smaller

Answer 1

To prove that the expressions \((x - 5)(x + 3)\) and \(2x^2 - x^2 - 2x - 20 + 5\) are equivalent, we'll expand Expression 1 and simplify Expression 2.

Expression 1: \((x - 5)(x + 3)\) Using the Distributive Property (FOIL method):

\[ (x - 5)(x + 3) = x\cdot x + x\cdot 3 - 5\cdot x - 5\cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] \[ = x^2 - 2x - 15 \]

Expression 2: \(2x^2 - x^2 - 2x - 20 + 5\) Now, simplify Expression 2 by combining like terms:

\[ 2x^2 - x^2 = x^2 \] \[ -2x \] \[ -20 + 5 = -15 \] Thus, we have: \[ \text{Expression 2} = x^2 - 2x - 15 \]

Since both expressions simplify to \(x^2 - 2x - 15\), we conclude that they are equivalent.