The problem is to classify linear expressions as either "Not Factored Form" or "Factored Form". There are six expressions to categorize, with a hint that there are 3 expressions in each category.

1. Problem Description

The problem is to classify linear expressions as either "Not Factored Form" or "Factored Form". There are six expressions to categorize, with a hint that there are 3 expressions in each category.

2. Solution Steps

We need to check each expression to see if the factoring is correct.

* Expression 1: $4x - 30 = 2(2x - 15)$

Distribute the 2 on the right side: $2(2x - 15) = 4x - 30$. This is correctly factored. So it belongs to the "Factored Form" category.

* Expression 2: $2x + 8 = 2(x + 4)$

Distribute the 2 on the right side: $2(x + 4) = 2x + 8$. This is correctly factored. So it belongs to the "Factored Form" category.

* Expression 3: $3x + 15 = 3(3x + 5)$

Distribute the 3 on the right side: $3(3x + 5) = 9x + 15$. This is NOT equal to $3x + 15$. So it belongs to the "Not Factored Form" category.

* Expression 4: $10 + 15x = 5(2 + 3x)$

Distribute the 5 on the right side: $5(2 + 3x) = 10 + 15x$. This is correctly factored. So it belongs to the "Factored Form" category.

* Expression 5: $24 - 4x = 2(12 - 2x)$

Distribute the 2 on the right side: $2(12 - 2x) = 24 - 4x$. This is correctly factored. So it belongs to the "Factored Form" category.

* Expression 6: $12 - 8x = 4(2 - 3x)$

Distribute the 4 on the right side: $4(2 - 3x) = 8 - 12x$. This is NOT equal to $12 - 8x$. So it belongs to the "Not Factored Form" category.

* We have now found four "Factored Form" options. Since there are meant to be only three, there must be an error. We will re-evaluate.

* Expression 1: $4x - 30 = 2(2x - 15)$. The right side expands to $4x - 30$. Correct.

* Expression 2: $2x + 8 = 2(x + 4)$. The right side expands to $2x + 8$. Correct.

* Expression 3: $3x + 15 = 3(3x + 5)$. The right side expands to $9x + 15$. Incorrect.

* Expression 4: $10 + 15x = 5(2 + 3x)$. The right side expands to $10 + 15x$. Correct.

* Expression 5: $24 - 4x = 2(12 - 2x)$. The right side expands to $24 - 4x$. Correct.

* Expression 6: $12 - 8x = 4(2 - 3x)$. The right side expands to $8 - 12x$. Incorrect.

Something is wrong, Expression 5 is correct. The mistake must be in expression

1. Expression 1: $4x-30 = 2(2x-15)$, the right side is: $2(2x-15)=4x-30$, which is factored correctly! Expression 1 is Factored Form.

Expression 6: $12-8x = 4(2-3x)$, we have $4(2-3x)=8-12x$, this is Not Factored Form.

We will now re-evaluate.

Since there are meant to be only three options for each type. And we already have determined what they are, there must be an error in this.

Looking closer we have an error in Expression 6, it must be: $12 - 8x = 4(3-2x)$, but it says $4(2-3x)$. Thus, we have the correct factorization. Now we must analyze it correctly.

Expression 6: $12 - 8x = 4(2 - 3x)$. This expands to $8 - 12x$. NOT EQUAL!

Expression 5: $24 - 4x = 2(12 - 2x)$. This expands to $24 - 4x$. CORRECT!

Expression 4: $10 + 15x = 5(2 + 3x)$. This expands to $10 + 15x$. CORRECT!

Expression 2: $2x + 8 = 2(x + 4)$. This expands to $2x + 8$. CORRECT!

The mistake is Expression 1, because $4x-30$ should have the $2$ factored out, and it would be $2(2x-15)$, so it is Factored Form.

The final check: Expression 6: $12 - 8x = 4(2 - 3x)$. NOT EQUAL!

Expression 3: $3x + 15 = 3(3x + 5)$. NOT EQUAL!

Here is the breakdown:

Not Factored Form:

$3x + 15 = 3(3x + 5)$

$12 - 8x = 4(2 - 3x)$

The correct answer is $4x-30=2(2x-15)$, $2x+8 = 2(x+4)$, and $10+15x=5(2+3x)$.

Expression 4: $10 + 15x = 5(2 + 3x)$ and so we have $5(2 + 3x) = 10 + 15x$, and we are left with, Expression 5: $24-4x=2(12-2x)$. Then this means $2(12-2x)=24-4x$. And lastly Expression 2: $2x+8 = 2(x+4)$, then $2(x+4)=2x+8$.

Final re-evaluation

Factor Form: $4x - 30 = 2(2x-15)$, $2x + 8 = 2(x+4)$, $10 + 15x = 5(2+3x)$

Not Factor Form: $3x + 15 = 3(3x + 5)$, $24 - 4x = 2(12 - 2x)$ is wrong, $12 - 8x = 4(2 - 3x)$.

Expression 5: $24-4x=2(12-2x)$ This SHOULD NOT BE in factored form and it should be: $24-4x=4(6-x)$.

$4x-30 = 2(2x-15)$ is FACTORED. $2x+8 = 2(x+4)$ is FACTORED. $10 + 15x = 5(2 + 3x)$ is FACTORED.

Then we have the following in NOT FACTORED form:

$3x + 15 = 3(3x + 5)$, and $24 - 4x = 2(12 - 2x)$ is wrong. But $12 - 8x = 4(2 - 3x)$.

3. Final Answer

Not Factored Form:

$3x + 15 = 3(3x + 5)$

$24 - 4x = 2(12-2x)$

$12 - 8x = 4(2 - 3x)$

Factored Form:

$4x - 30 = 2(2x - 15)$

$2x + 8 = 2(x + 4)$

$10 + 15x = 5(2 + 3x)$