Math Problem Statement

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

Solution

The given expression to simplify and factor is:

(a+b+2)(a+b)(ab)2+1(a + b + 2)(a + b) - (a - b)^2 + 1

Step 1: Expand the terms

Expand both parts separately:

  • First term: (a+b+2)(a+b)(a + b + 2)(a + b) Using distributive property: (a+b+2)(a+b)=(a+b)2+2(a+b)(a + b + 2)(a + b) = (a + b)^2 + 2(a + b) Expanding further: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 and 2(a+b)=2a+2b2(a + b) = 2a + 2b So: (a+b+2)(a+b)=a2+2ab+b2+2a+2b(a + b + 2)(a + b) = a^2 + 2ab + b^2 + 2a + 2b

  • Second term: (ab)2-(a - b)^2 Expanding (ab)2(a - b)^2: (ab)2=a22ab+b2(a - b)^2 = a^2 - 2ab + b^2 So: (ab)2=(a22ab+b2)=a2+2abb2-(a - b)^2 = -(a^2 - 2ab + b^2) = -a^2 + 2ab - b^2

Step 2: Combine the expressions

Now, substitute back the expanded terms into the original expression: a2+2ab+b2+2a+2ba2+2abb2+1a^2 + 2ab + b^2 + 2a + 2b - a^2 + 2ab - b^2 + 1 Combine like terms:

  • a2a^2 cancels out: a2a2=0a^2 - a^2 = 0
  • b2b^2 cancels out: b2b2=0b^2 - b^2 = 0
  • Combine 2ab+2ab=4ab2ab + 2ab = 4ab
  • The remaining terms: 2a+2b+12a + 2b + 1

So, the simplified expression is: 4ab+2a+2b+14ab + 2a + 2b + 1

Step 3: Factor the expression

Now factor the simplified expression: 4ab+2a+2b+1=(2a+1)(2b+1)4ab + 2a + 2b + 1 = (2a + 1)(2b + 1)

Conclusion:

The correct answer is: E:(2b+1)(2a+1)\boxed{E: (2b + 1)(2a + 1)}

Would you like more details or have any further questions?

Related Questions:

  1. How do you factor quadratic expressions like a2+2ab+b2a^2 + 2ab + b^2?
  2. What are the steps to expand binomial expressions?
  3. How do you recognize common factoring patterns in expressions?
  4. What is the distributive property and how is it used in algebra?
  5. How can you solve similar factorization problems with three variables?

Tip:

Always double-check by re-expanding your factored terms to verify they match the original expression.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Simplifying Expressions
Quadratic Expansions

Formulas

Distributive property: (a + b)(c + d) = ac + ad + bc + bd
(a - b)^2 = a^2 - 2ab + b^2

Theorems

Distributive Property
Factoring and Expanding Binomials

Suitable Grade Level

Grades 9-11