Final answer:
The factorization of the polynomial 1 - 2a - 2b - 3(a + b)^2 is achieved by first expanding the (a+b)^2 term, and then distributing the -3 into that expression. This results in the expression 1 - 2a - 2b - 3a^2 -6ab -3b^2, which cannot be further factorized.
Step-by-step explanation:
The subject of the question is Mathematics, specifically algebra, and the topic is factoring a polynomial.
To factorize the expression 1 - 2a - 2b - 3(a + b)^2, we first expand the term (a+b)^2 and then distribute the -3 into the resulting expression.
Step 1: (a+b)^2 = a^2 + 2ab + b^2
Step 2: We substitute (a+b)^2 with the result from step 1 and distribute -3 across the expression:
Now the original expression 1 - 2a - 2b - 3(a + b)^2 becomes 1 - 2a - 2b - 3(a^2 + 2ab + b^2) which equals 1 - 2a - 2b - 3a^2 -6ab -3b^2.
This polynomial cannot be further factorized, hence the factorization of the original polynomial 1 - 2a - 2b - 3(a + b)^2 is 1 - 2a - 2b - 3a^2 -6ab -3b^2.
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