Let's simplify the expression [tex]\( 4 \sqrt{7} - 3x \sqrt{7} - x \sqrt{7} \)[/tex] step-by-step.
1. First, we observe the given expression:
[tex]\[
4 \sqrt{7} - 3x \sqrt{7} - x \sqrt{7}
\][/tex]
2. Notice that each term contains a common factor of [tex]\( \sqrt{7} \)[/tex]. So, we can factor out [tex]\( \sqrt{7} \)[/tex] from the expression:
[tex]\[
\sqrt{7} \left( 4 - 3x - x \right)
\][/tex]
3. Next, we simplify the expression inside the parentheses:
[tex]\[
4 - 3x - x = 4 - 4x
\][/tex]
4. Substituting this back into our factored form, we get:
[tex]\[
\sqrt{7} (4 - 4x)
\][/tex]
5. Distribute [tex]\(\sqrt{7}\)[/tex] back into the simplified expression:
[tex]\[
\sqrt{7} (4 - 4x) = 4\sqrt{7} (1 - x)
\][/tex]
So, we see that the simplified form of the expression [tex]\( 4 \sqrt{7} - 3x \sqrt{7} - x \sqrt{7} \)[/tex] is [tex]\( 4 \sqrt{7} (1 - x) \)[/tex].
Looking at the given choices:
- A: [tex]\( 4 \sqrt{7} - 4 x \sqrt{7} \)[/tex]
- B: [tex]\( -x^2 \)[/tex]
- C: [tex]\( 0 \)[/tex]
- D: [tex]\( -2 x \sqrt{7} \)[/tex]
The simplified expression [tex]\( 4 \sqrt{7} (1 - x) \)[/tex] matches choice A.
Therefore, the correct answer is:
[tex]\[
\boxed{4 \sqrt{7} - 4 x \sqrt{7}}
\][/tex]
