Factorization by pattern recognition: Factor the expression a^2 + 4b^2 + 4b − 4ab − 2a − 8 into a product of two linear factors.
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A(a − 2b − 4)(a − 2b + 2)
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B(a − b + 2)(a + 4b + 4)
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C(a + 2b − 4)(a + 2b + 2)
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D(a + 2b − 1)(a − 2b + 1)
Answer
Correct Answer: (a − 2b − 4)(a − 2b + 2)
Explanation
Introduction / Context:We aim to factor a bivariate quadratic expression. Grouping terms or using a substitution like x = a − 2b to identify a perfect quadratic in x can simplify the process drastically, turning it into a one-variable factorization task.Given Data / Assumptions:
- Expression: a^2 + 4b^2 + 4b − 4ab − 2a − 8.
- Real a, b.
Concept / Approach:Note that (a − 2b)^2 = a^2 − 4ab + 4b^2 appears within the expression. Set x = a − 2b and rewrite the expression in terms of x, then factor the resulting quadratic in x.
Step-by-Step Solution:
Rewrite: a^2 + 4b^2 − 4ab = (a − 2b)^2.Thus the expression becomes (a − 2b)^2 − 2a + 4b − 8.But a = (a − 2b) + 2b, so −2a + 4b = −2(a − 2b) − 0.Let x = a − 2b. Then expression = x^2 − 2x − 8 = (x − 4)(x + 2).Substitute back: (a − 2b − 4)(a − 2b + 2).Verification / Alternative check:Expand (a − 2b − 4)(a − 2b + 2) = (a − 2b)^2 − 2(a − 2b) − 8 = a^2 + 4b^2 − 4ab − 2a + 4b − 8, matching the original expression exactly.
Why Other Options Are Wrong:
- Other pairs expand to different cross terms (e.g., +4ab instead of −4ab, or wrong linear coefficients), not reproducing the original expression.
Common Pitfalls:Incorrect grouping leading to sign errors, or choosing x = a + 2b (which yields the wrong middle terms). The substitution x = a − 2b fits the pattern perfectly.
Final Answer:
(a − 2b − 4)(a − 2b + 2)