Compare Expressions: Finding Equivalence in (6b+3)(a-2) and Related Forms

Which of the following expressions have the same value?

1. $(6b+3)(-2+a)$

2. $(2b+1)(3a-6)$

3. $(a+3)(6b-2)$

4. $6ab+3a-12b-6$

To solve this problem, we need to systematically expand and simplify each expression given in the problem statement:

• Expression 1: $(6b+3)(-2+a)$

  Expand using the distributive property:

  $= 6b(-2) + 6b(a) + 3(-2) + 3(a)$

  $= -12b + 6ab - 6 + 3a$

  Reorder terms: $6ab + 3a - 12b - 6$

• Expression 2: $(2b+1)(3a-6)$

  Expand using the distributive property:

  $= 2b(3a) + 2b(-6) + 1(3a) + 1(-6)$

  $= 6ab - 12b + 3a - 6$

  This simplifies directly to $6ab + 3a - 12b - 6$

• Expression 3: $(a+3)(6b-2)$

  Expand using the distributive property:

  $= a(6b) + a(-2) + 3(6b) + 3(-2)$

  $= 6ab - 2a + 18b - 6$

  This results in $6ab - 2a + 18b - 6$, clearly different from the others

• Expression 4: $6ab+3a-12b-6$

  This is already simplified and the same as the results of expressions 1 and 2.

Upon comparing the simplified expressions, expressions 1, 2, and 4 have the same value: $6ab + 3a - 12b - 6$. Expression 3 differs with $6ab - 2a + 18b - 6$.

Thus, the expressions with the same value are 1, 2, and 4.

Therefore, the correct answer is choice 4: $1,2,4$.