Sure, let's solve the given expression step-by-step and determine which option is correct.
We start with the expression:
[tex]\[
\frac{x^6 - x}{4y}
\][/tex]
Given [tex]\( x = -4 \)[/tex] and [tex]\( y = 4 \)[/tex], we first evaluate [tex]\( x^6 - x \)[/tex]:
1. Calculate [tex]\( x^6 \)[/tex]:
[tex]\[
(-4)^6 = 4096
\][/tex]
2. Subtract [tex]\( x \)[/tex] from [tex]\( x^6 \)[/tex]:
[tex]\[
4096 - (-4) = 4096 + 4 = 4100
\][/tex]
Next, we substitute [tex]\( y = 4 \)[/tex] into the denominator:
[tex]\[
4y = 4 \cdot 4 = 16
\][/tex]
Now, we can write the reduced expression:
[tex]\[
\frac{4100}{16}
\][/tex]
To simplify this fraction:
[tex]\[
\frac{4100}{16} = 256.25
\][/tex]
Now let's compare this result to the given options:
A. [tex]\(\frac{1025}{4} = 256.25\)[/tex]
B. [tex]\(\frac{1023}{4} = 255.75\)[/tex]
C. [tex]\(\frac{1023}{4} = 255.75\)[/tex]
D. [tex]\(\frac{16385}{4} = 4096.25\)[/tex]
From the options, option A, [tex]\(\frac{1025}{4}\)[/tex], equals 256.25, which matches our result.
Thus, the correct answer is:
[tex]\[
\boxed{\frac{1025}{4}}
\][/tex]
