Factorization by pattern recognition: Factor the expression a^2 + 4b^2 + 4b − 4ab − 2a − 8 into a product of two linear factors.

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:

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: