Finding Equivalent Forms of 12n+48-36mn-144m: Algebraic Expression Analysis

Which of the expressions are equal to the expression?

$12n+48-36mn-144m$

1. $12(1-3m)(n+4)$

2. $3(4-12m)(n+4)$

3. $12(-3m+4)(n+1)$

4. $12(n+4)-3m(12n+48)$

Let's start by factoring the given expression $12n + 48 - 36mn - 144m$.

First, notice that:

• In the first two terms: $12n + 48$, we can factor out $12$, giving us $12(n + 4)$.

• In the last two terms: $-36mn - 144m$, we can factor out $-36m$, resulting in $-36m(n + 4)$.

Combining both factorizations, we can write the original expression as:

$12(n + 4) - 36m(n + 4)$

Now, we factor out the common term $(n + 4)$:

$(12 - 36m)(n + 4)$

Thus, the expression simplifies to:

$12(1 - 3m)(n + 4)$

Now, let's verify which options match:

Option 1: $12(1-3m)(n+4)$
This directly matches our simplified expression, so it is a correct choice.

Option 2: $3(4 - 12m)(n + 4)$
Simplifying: Factoring 3 from $4 - 12m$ gives $3 \times 4(1 - 3m)$, which matches the expression $12(1 - 3m)(n + 4)$. So, this is also a correct choice.

Option 3: $12(-3m + 4)(n + 1)$
The factors do not align with our expression because $(n+1)$ is not factored from $(n + 4)$.

Option 4: $12(n + 4) - 3m(12n + 48)$
Rewriting: $12(n+4) - 3m(12(n+4)) = (12 - 36m)(n+4)$ which matches. Therefore, it is correct.

Hence, options 1, 2, and 4 are equivalent to the original expression.

The correct answer to the problem is

$1, 2, 4$