To determine the sign of the expression [tex]\(-\frac{a^4}{3}\)[/tex] when [tex]\(a < 0\)[/tex], let's analyze it step by step:
1. Consider the value of [tex]\(a\)[/tex]:
Since [tex]\(a < 0\)[/tex], [tex]\(a\)[/tex] is a negative number.
2. Calculate [tex]\(a^4\)[/tex]:
When a negative number is raised to an even power, the result is positive. Hence, [tex]\(a^4\)[/tex] will be a positive number.
For example, if [tex]\(a = -1\)[/tex], then:
[tex]\[
a^4 = (-1)^4 = 1
\][/tex]
3. Divide [tex]\(a^4\)[/tex] by 3:
Dividing a positive number by 3 still results in a positive number.
Continuing the example with [tex]\(a = -1\)[/tex]:
[tex]\[
\frac{a^4}{3} = \frac{1}{3} > 0
\][/tex]
4. Multiply by [tex]\(-1\)[/tex]:
Finally, we need to consider the negative sign in front of the fraction. Multiplying a positive number by [tex]\(-1\)[/tex] results in a negative number.
Thus:
[tex]\[
-\frac{a^4}{3} = - \left(\frac{1}{3}\right) < 0
\][/tex]
Therefore, regardless of the specific value of [tex]\(a\)[/tex] (as long as [tex]\(a < 0\)[/tex]), the expression [tex]\(-\frac{a^4}{3}\)[/tex] will always be negative.
Conclusion:
The correct answer is [tex]\(\boxed{\text{B) Negative}}\)[/tex].
