In Exercises 69–80, factor completely.(x + y)² + 6(x + y) + 9

Factoring quadratic expressions involves rewriting them as a product of binomials. This process is essential for simplifying expressions and solving equations. The standard form of a quadratic is ax² + bx + c, and the goal is to express it in the form (px + q)(rx + s). Recognizing patterns, such as perfect squares or the difference of squares, can aid in this process.

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique involves manipulating the expression to create a squared term, which can simplify factoring. For example, in the expression (x + y)² + 6(x + y) + 9, recognizing that it can be rewritten as ((x + y) + 3)² helps in identifying the factors.

The Binomial Theorem provides a formula for expanding expressions raised to a power, specifically (a + b)ⁿ. In the context of factoring, it helps in recognizing patterns in polynomials. For instance, the expression (x + y)² can be expanded to x² + 2xy + y², which is useful for identifying and factoring quadratic expressions effectively.