To prove that the expressions \( (x - 7)(x + 4) \) (Expression 1) and \( x^2 - 3x - 28 \) (Expression 2) are equivalent, we can use the Distributive Property step by step.
### Expression 1: \( (x - 7)(x + 4) \)
We start with Expression 1, which is \( (x - 7)(x + 4) \). Using the Distributive Property, we multiply each term in the first parentheses by each term in the second parentheses. First, we take \( x \) from the first parentheses and distribute it to both terms in the second parentheses, giving us \( x \cdot x + x \cdot 4 = x^2 + 4x \). Then, we take \(-7\) from the first parentheses and distribute it to both terms in the second parentheses, resulting in \(-7 \cdot x - 7 \cdot 4 = -7x - 28\).
Now we combine all the terms together: \( x^2 + 4x - 7x - 28 \). After combining the like terms \( 4x \) and \(-7x\), we get \( x^2 - 3x - 28 \).
### Expression 2: \( x^2 - 3x - 28 \)
Now let’s look at Expression 2, which is already written as \( x^2 - 3x - 28 \). This expression clearly shows the terms with \( x^2 \), the linear term \(-3x\), and the constant term \(-28\). We can see that this expression is the expanded result from Expression 1 after applying the Distributive Property.
Since we have shown that expanding Expression 1 leads us precisely to the form in Expression 2, we have proven that \( (x - 7)(x + 4) \) is equivalent to \( x^2 - 3x - 28 \). This confirms that both expressions represent the same polynomial.