The given expression is $(x - 5)(x + 7)$, and we need to find which of the provided options is equivalent to it.

We will expand the expression using the distributive property (also known as FOIL in this case):

$(x - 5)(x + 7) = x(x + 7) - 5(x + 7)$

Expanding both terms:

$x(x + 7) = x^2 + 7x$ $-5(x + 7) = -5x - 35$

Now, combine the like terms:

$x^2 + 7x - 5x - 35 = x^2 + 2x - 35$

Thus, the equivalent expression is:

$x^2 + 2x - 35$

The correct answer is A: $x^2 + 2x - 35$.

Would you like further details or have any questions? Here are five related questions to expand your understanding:

1. How would you expand a product of two binomials like $(x + a)(x + b)$?
2. What is the distributive property, and why is it useful in algebra?
3. Can you explain how to identify and combine like terms in polynomial expressions?
4. What happens when the terms inside the binomials contain variables other than $x$?
5. What is the difference between the FOIL method and regular distribution?

Tip: When multiplying binomials, always ensure you combine like terms after expanding to simplify the expression fully.