(x+2)^2+5(x+2)+4

Jasonbradley

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Jun 16, 2024

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Factoring the Quadratic Expression: (x+2)^2 + 5(x+2) + 4

This article will explore the process of factoring the quadratic expression: (x+2)^2 + 5(x+2) + 4. We will use a technique known as substitution to simplify the expression and make it easier to factor.

Substitution Method

1. Identify the Repeated Term: Observe that the expression contains the term (x+2) repeated multiple times.
2. Introduce a Substitute: Let's substitute y = (x+2). This will simplify our expression.
3. Rewrite the Expression: Now, our expression becomes: y^2 + 5y + 4.

Factoring the Simplified Expression

The simplified expression is now a standard quadratic equation in terms of 'y'. We can factor this expression using the following steps:

1. Find Two Numbers: We need to find two numbers that:
   • Multiply to give 4 (the constant term).
   • Add up to give 5 (the coefficient of the 'y' term).
2. Identify the Numbers: The two numbers that satisfy these conditions are 4 and 1.
3. Factor the Expression: We can now factor the expression as: (y + 4)(y + 1).

Back Substitution

1. Replace 'y' with (x+2): Now, we substitute back y = (x+2) into the factored expression: (x+2+4)(x+2+1).
2. Simplify: This simplifies to (x+6)(x+3).

Final Result

Therefore, the factored form of the expression (x+2)^2 + 5(x+2) + 4 is (x+6)(x+3).