Answer :
Let's go through Kate's steps to identify where she made a mistake.
### Step 1
Kate needs to apply the distributive property to the binomials [tex]\((x - 2)(3x + 4)\)[/tex]. She does this correctly in the first step:
[tex]\[ (x-2)(3x) + (x-2)(4) \][/tex]
This step is correct.
### Step 2
Next, she needs to distribute each term in the first binomial to each term in the second binomial:
[tex]\[ (x)(3x) + (-2)(3x) + (x)(4) + (-2)(4) \][/tex]
Breaking it down:
[tex]\[ 3x^2 + (-6x) + 4x + (-8) \][/tex]
This step is also correct.
### Step 3
Kate combines the like terms:
[tex]\[ 3x^2 + (-6x) + 4x - 8 \][/tex]
Simplifying further, she should have combined the [tex]\( -6x \)[/tex] and [tex]\( 4x \)[/tex]:
[tex]\[ 3x^2 - 2x - 8 \][/tex]
So she should have had:
3x^2 + 4x - 6x - 8 = 3x^2 - 2x - 8
This step is correct as well.
### Step 4
However, examining her final result:
[tex]\[ 3x^2 + 10x - 8 \][/tex]
Shows that she incorrectly combined the terms 4x and -6x that result should be -2x, but she wrote +10x instead. Therefore, the first error appears in:
[tex]\[ \boxed{\text{Step 4}} \][/tex]
The correct answer is C. Step 4.
### Step 1
Kate needs to apply the distributive property to the binomials [tex]\((x - 2)(3x + 4)\)[/tex]. She does this correctly in the first step:
[tex]\[ (x-2)(3x) + (x-2)(4) \][/tex]
This step is correct.
### Step 2
Next, she needs to distribute each term in the first binomial to each term in the second binomial:
[tex]\[ (x)(3x) + (-2)(3x) + (x)(4) + (-2)(4) \][/tex]
Breaking it down:
[tex]\[ 3x^2 + (-6x) + 4x + (-8) \][/tex]
This step is also correct.
### Step 3
Kate combines the like terms:
[tex]\[ 3x^2 + (-6x) + 4x - 8 \][/tex]
Simplifying further, she should have combined the [tex]\( -6x \)[/tex] and [tex]\( 4x \)[/tex]:
[tex]\[ 3x^2 - 2x - 8 \][/tex]
So she should have had:
3x^2 + 4x - 6x - 8 = 3x^2 - 2x - 8
This step is correct as well.
### Step 4
However, examining her final result:
[tex]\[ 3x^2 + 10x - 8 \][/tex]
Shows that she incorrectly combined the terms 4x and -6x that result should be -2x, but she wrote +10x instead. Therefore, the first error appears in:
[tex]\[ \boxed{\text{Step 4}} \][/tex]
The correct answer is C. Step 4.
