In Exercises 1–68, factor completely, or state that the polynomial is prime. (x + 5)(x − 3) + (x + 5)(x − 7)

Factoring polynomials involves expressing a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions and solving equations. In the given question, recognizing common factors in the polynomial will help in breaking it down into simpler forms.

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by two or more terms inside parentheses. This property is crucial for expanding expressions and simplifying polynomials, as seen in the initial step of the problem where we need to combine like terms after applying this property.

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is vital in polynomial manipulation, as it helps in reducing the expression to a more manageable form before factoring or determining if the polynomial is prime.