To determine if \((x - 5)(x + 3)\) is equivalent to \(2x^2 - x^2 - 2x - 20 + 5\), we will first simplify each expression.
- Simplifying \((x - 5)(x + 3)\) using the distributive property (also known as the FOIL method for binomials):
\[ (x - 5)(x + 3) = x \cdot x + x \cdot 3 - 5 \cdot x - 5 \cdot 3 \] \[ = x^2 + 3x - 5x - 15 \] Combine like terms: \[ = x^2 - 2x - 15 \]
- Simplifying \(2x^2 - x^2 - 2x - 20 + 5\):
First, combine like terms: \[ 2x^2 - x^2 = x^2 \] Now combine \(-20 + 5\): \[ -20 + 5 = -15 \] Putting it all together: \[ x^2 - 2x - 15 \]
Now we have both simplified expressions:
- From \((x - 5)(x + 3)\), we get \(x^2 - 2x - 15\).
- From \(2x^2 - x^2 - 2x - 20 + 5\), we also get \(x^2 - 2x - 15\).
Since both expressions simplify to \(x^2 - 2x - 15\), we can conclude that:
\[ (x - 5)(x + 3) \text{ is equivalent to } 2x^2 - x^2 - 2x - 20 + 5. \]