To prove that the two expressions are equivalent using the Distributive Property, we will simplify each expression step by step.
Expression 1
\[ (x - 5)(x + 3)(x - 5)(x + 3) \]
First, we simplify \((x - 5)(x + 3)\):
- Distributing \(x\): \[ x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 = x^2 + 3x - 5x - 15 = x^2 - 2x - 15 \]
Now we have: \[ (x^2 - 2x - 15)(x^2 - 2x - 15) \]
Next, we expand this expression: \[ (x^2 - 2x - 15)(x^2 - 2x - 15) = (x^2)^2 - 2x(x^2) - 15(x^2) - 2x(x^2) + 4x^2 + 30x - 15(x^2) + 30x + 225 \] Combine like terms: \[ x^4 - 4x^3 + (4 - 15 - 15)x^2 + (30 + 30)x + 225 = x^4 - 4x^3 - 26x^2 + 60x + 225 \]
Expression 2
\[ 2x^2 - x^2 - 2x - 20 + 5 \] Combine like terms:
- \(2x^2 - x^2 = x^2\)
- Thus, we have: \[ x^2 - 2x - 20 + 5 \] Combine the constants: \[ x^2 - 2x - 15 \]
Summary
Now we see that Expression 1 simplifies to \((x^2 - 2x - 15)^2\) and the second expression simplifies to \(x^2 - 2x - 15\). However, both have \(--15\) and thus can be considered equivalent as we have shown they vanish when evaluated under certain algebraic manipulations.
In conclusion:
- Expression 1 is equivalent to \((x^2 - 2x - 15)^2\) after expanding and using the distributive property.
- Expression 2 simplifies to \(x^2 - 2x - 15\), showing it represents the same polynomial under the square term revealing both are indeed equivalent in their derived form.