Factor each trinomial, if possible. See Examples 3 and 4. (2p+q)²-10(2p+q)+25

Verified step by step guidance

1. Recognize that the expression is a quadratic in terms of the binomial \( (2p + q) \). Let \( x = (2p + q) \) to simplify the expression.
2. Rewrite the expression using \( x \): \( x^2 - 10x + 25 \). This is now a standard quadratic trinomial.
3. Look for two numbers that multiply to the constant term (25) and add up to the coefficient of \( x \) (-10). These numbers are both -5 because \( -5 \times -5 = 25 \) and \( -5 + -5 = -10 \).
4. Use these numbers to factor the quadratic as a perfect square trinomial: \( (x - 5)^2 \).
5. Substitute back \( x = (2p + q) \) to get the factored form: \( ((2p + q) - 5)^2 \).

Key Concepts

Recognizing Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically in the form ax^2 + bx + c. In this problem, the expression is written in terms of a binomial (2p + q), which can be treated as a single variable to simplify factoring.

Factoring Perfect Square Trinomials

A perfect square trinomial takes the form (a ± b)^2 = a^2 ± 2ab + b^2. Identifying this pattern helps factor expressions quickly. Here, the trinomial resembles (2p + q)^2 - 10(2p + q) + 25, which can be seen as (2p + q - 5)^2.

Substitution Method in Factoring

Substitution involves temporarily replacing a complex expression with a single variable to simplify the factoring process. By letting x = (2p + q), the trinomial becomes x^2 - 10x + 25, which is easier to factor before substituting back.