In Exercises 83–92, factor by introducing an appropriate substitution.3(x+1)² − 5(x+1) + 2 (Let u = x+1)

Verified step by step guidance

1. Identify the substitution: Let \( u = x + 1 \).
2. Rewrite the expression using \( u \): \( 3u^2 - 5u + 2 \).
3. Recognize that this is a quadratic expression in terms of \( u \).
4. Factor the quadratic expression \( 3u^2 - 5u + 2 \) using methods such as factoring by grouping or the quadratic formula.
5. Once factored, substitute back \( u = x + 1 \) into the factored expression to express the solution in terms of \( x \).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. In the given question, recognizing the structure of the polynomial allows for easier manipulation and solution.

Substitution Method

The substitution method is a technique used to simplify complex expressions by replacing a variable or expression with a single variable. In this case, letting u = x + 1 transforms the polynomial into a more manageable form, making it easier to factor and solve.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically in the form ax² + bx + c. They can often be factored into the product of two binomials. Understanding the characteristics of quadratic expressions is crucial for applying factoring techniques effectively, as seen in the exercise.