(2a+b)^2-6a-3b-4

2 min read Jun 16, 2024

Factoring the Expression (2a + b)^2 - 6a - 3b - 4

This article explores the process of factoring the algebraic expression (2a + b)^2 - 6a - 3b - 4. We will break down the steps and use different techniques to simplify the expression.

Step 1: Expanding the Square

First, we expand the square term:

(2a + b)^2 = (2a + b)(2a + b) = 4a^2 + 4ab + b^2

Now, the expression becomes: 4a^2 + 4ab + b^2 - 6a - 3b - 4

Step 2: Rearranging Terms

To facilitate factoring, let's rearrange the terms by grouping similar terms together:

4a^2 - 6a + 4ab - 3b + b^2 - 4

Step 3: Factoring by Grouping

We can factor by grouping the first two, the next two, and the last two terms:

2a(2a - 3) + b(4a - 3) + (b^2 - 4)

Notice that we now have a common factor of (2a - 3) in the first two groups. Additionally, the last group is a difference of squares:

(2a - 3)(2a + b) + (b - 2)(b + 2)

Step 4: Final Factorization

Finally, we can factor out the common factor (2a - 3):

(2a - 3)(2a + b + b - 2)

Simplifying the expression:

(2a - 3)(2a + 2b - 2)

We can factor out a 2 from the second factor:

(2a - 3)(2)(a + b - 1)

Therefore, the factored form of the expression (2a + b)^2 - 6a - 3b - 4 is 2(2a - 3)(a + b - 1).