Factor by grouping.
36 + ab + 2b + 18a
A. (a-2)(b-18)
B. (a+2)(b+18)
C. (a+2)(18a+2)
D. (ab+2b)(36+18a)
E. this cannot be factored.
b(a+2)+18(2+a)
=(b+18)(a+2)
so B
To factor by grouping, you need to group terms that have common factors. Let's examine the given expression: 36 + ab + 2b + 18a.
Step 1: Group the terms.
Group the first two terms and the last two terms: (36 + ab) + (2b + 18a).
Step 2: Look for the common factors.
In the first group, both terms have no common factors. In the second group, both terms have a common factor of 2.
Step 3: Factor out the common factors.
In the first group, we cannot factor out a common factor.
In the second group, factor out 2 to get: 2(b + 9a).
Now, we have: 36 + ab + 2(b + 9a).
Step 4: Look for the common factors in the entire expression.
In the entire expression, there is no common factor other than 1.
Therefore, the final factored form by grouping is: 36 + ab + 2(b + 9a).
Now, let's compare this expression with the answer choices:
A. (a-2)(b-18): This is not equivalent to the given expression.
B. (a+2)(b+18): This is not equivalent to the given expression.
C. (a+2)(18a+2): This is not equivalent to the given expression.
D. (ab+2b)(36+18a): This is not equivalent to the given expression.
E. This cannot be factored: This is not true since we have successfully factored the given expression.
Therefore, the correct answer is E. This expression can be factored as 36 + ab + 2(b + 9a).