Let's carefully analyze Julian's steps:
### Original Expression
[tex]\[ 2x^4 + 2x^3 - x^2 - x \][/tex]
### Step 1:
Julian factored out [tex]\( x \)[/tex] from the original expression:
[tex]\[ 2x^4 + 2x^3 - x^2 - x = x \cdot (2x^3 + 2x^2 - x - 1) \][/tex]
This step is correct.
### Step 2:
Next, Julian attempted to factor by grouping:
[tex]\[ x \cdot (2x^3 + 2x^2 - x - 1) = x \left[ 2x^2(x + 1) - 1(x - 1) \right] \][/tex]
Here we should check his grouping and factoring.
- Group the terms: [tex]\( (2x^3 + 2x^2) \)[/tex] and [tex]\( (-x - 1) \)[/tex]
- Factor [tex]\( 2x^2 \)[/tex] from the first group: [tex]\( 2x^2(x + 1) \)[/tex]
- Factor [tex]\( -1 \)[/tex] from the second group: [tex]\( -1(x - 1) \)[/tex]
Julian's mistake occurs here. When factoring the second group, he didn't apply the distributive property correctly:
[tex]\[ -1(x - 1) \neq 1(x - 1) \][/tex]
It should be:
[tex]\[ -1(x + 1 - 2) = -1(x + 1) - 2 \][/tex]
### Step 3:
Julian attempted to write the final factorized form:
[tex]\[ x \left(2x^2 - 1 \right)(x + 1)(x - 1) \][/tex]
Thus, Julian made his first mistake in Step 2. He incorrectly applied the distributive property when factoring out [tex]\(-1\)[/tex].
### Correct Step and Statement:
- Step 2 is where Julian made his first mistake.
- The mistake is described by Statement 2: Julian incorrectly applied the distributive property when factoring out -1.
Therefore, the correct step and statement are:
[tex]\[ \text{Step: 2, Statement: 2} \][/tex]
