In Exercises 39–44, factor by introducing an appropriate substitution.(x + 1)² + 8(x + 1) + 7 (Let u = x+1.)

Verified step by step guidance

1. Identify the substitution: Let \( u = x + 1 \).
2. Rewrite the expression using the substitution: \( u^2 + 8u + 7 \).
3. Recognize that this is a quadratic expression in terms of \( u \).
4. Factor the quadratic expression \( u^2 + 8u + 7 \) by finding two numbers that multiply to 7 and add to 8.
5. Substitute back \( u = x + 1 \) into the factored expression to express the solution in terms of \( x \).

Key Concepts

Substitution Method

The substitution method involves replacing a complex expression with a simpler variable to make calculations easier. In this case, letting u = x + 1 transforms the original expression into a quadratic form, simplifying the factoring process. This technique is particularly useful in polynomial equations where direct factoring may be cumbersome.

Factoring Quadratics

Factoring quadratics is the process of expressing a quadratic equation in the form ax² + bx + c as a product of two binomials. This is essential for solving equations or simplifying expressions. Understanding how to identify the coefficients and apply methods like the AC method or completing the square is crucial for effective factoring.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)². In the given expression, (x + 1)² represents a binomial that can be expanded to x² + 2x + 1. Recognizing this expansion helps in simplifying and rearranging terms for easier factoring.