Compare Expressions: Finding Matches for 2ab-4bc

Let's determine the equivalence of different expressions to $2ab-4bc$ by factorization:

Step 1: Factor the given expression:

The expression $2ab - 4bc$ has a common factor of $2b$.

Factor out $2b$, we obtain:

$\begin{aligned} 2ab - 4bc &= 2b(a) - 2b(2c) \\ &= 2b(a - 2c) \end{aligned}$

Step 2: Compare with each option:

• Option 1: $2b(a-2c)$

  • This is identical to the factorized form $2b(a-2c)$, so it is equivalent.

• Option 2: $2b(-2c+a)$

  • Although it appears reversed, $-2c + a$ is equivalent to $a - 2c$, so it's equivalent.

• Option 3: $2(-2bc+ab)$

  • By rearranging:

    $\begin{aligned} 2(-2bc + ab) &= 2(ab - 2bc) \\ &= 2ab - 4bc \end{aligned}$
    

  • It matches the original expression, thus is equivalent.

• Option 4: $2a(2bc-b)$

  • Expanding:

    $2a(2bc-b) = 4abc - 2ab$
    

  • Does not match $2ab - 4bc$, so not equivalent.

Conclusion: The expressions equivalent to $2ab - 4bc$ are Options 1, 2, and 3.

Therefore, the solution to the problem is $1,2,3$.