Factor each polynomial. See Example 7. 4(5x+7)²+12(5x+7)+9

Verified step by step guidance

1. Recognize that the polynomial is expressed in terms of the binomial \( (5x+7) \). To simplify the factoring process, let \( u = (5x+7) \). This substitution transforms the polynomial into a quadratic in terms of \( u \): \( 4u^2 + 12u + 9 \).
2. Identify the quadratic expression \( 4u^2 + 12u + 9 \) and prepare to factor it. Look for two numbers that multiply to \( 4 \times 9 = 36 \) and add up to \( 12 \).
3. Find the pair of numbers that satisfy the conditions: these numbers are \( 6 \) and \( 6 \) because \( 6 \times 6 = 36 \) and \( 6 + 6 = 12 \).
4. Rewrite the middle term \( 12u \) as \( 6u + 6u \) to facilitate factoring by grouping: \( 4u^2 + 6u + 6u + 9 \).
5. Group the terms and factor each group: \( (4u^2 + 6u) + (6u + 9) \). Factor out the greatest common factor from each group: \( 2u(2u + 3) + 3(2u + 3) \). Then, factor out the common binomial \( (2u + 3) \), resulting in \( (2u + 3)(2u + 3) \) or \( (2u + 3)^2 \). Finally, substitute back \( u = (5x + 7) \) to get the factored form \( (2(5x + 7) + 3)^2 \).

Key Concepts

Polynomial Factoring

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or expressions. This process helps simplify expressions and solve equations. Common methods include factoring out the greatest common factor, grouping, and recognizing special patterns like trinomials or perfect square trinomials.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial, such as (ax + b)^2 = a^2x^2 + 2abx + b^2. Recognizing this pattern allows for quick factoring and simplification of expressions involving squared binomials.

Substitution Method in Factoring

Substitution involves replacing a complex expression with a single variable to simplify factoring. For example, letting u = (5x + 7) transforms the polynomial into a quadratic in terms of u, making it easier to factor before substituting back the original expression.