A part of the sentence is underlined. Choose the alternative that improves the underlined part; if no improvement is needed, select "No improvement": After a hard days work, I just want to go home.

Introduction / Context:
Sentence improvement questions test the learner's command of grammar, usage, and idiomatic expressions in English. In this item, the focus is on correct possessive form and a very common phrase used in everyday speech and writing. Such questions are frequent in competitive exams because they quickly show whether a candidate notices small but important grammatical details like the placement of the apostrophe.

Given Data / Assumptions:
The sentence given is: After a hard days work, I just want to go home. The underlined part is hard days work, and the options suggest different versions: hard day's work, hard days' work, hard day work, and so on. The basic idea is that a day of work was hard, and we must choose the most natural and grammatically correct way to express this.

Concept / Approach:
In English, when one noun is used in a possessive sense before another noun, we often use the apostrophe s structure. The meaning here is the work of a hard day or the work belonging to a hard day. The standard expression used by native speakers is a hard day's work, where the singular noun day takes apostrophe s to show that the work is strongly associated with that day. The plural form days' would suggest many days, which does not match the phrase a hard day's work. Similarly, leaving out the apostrophe is incorrect because it fails to show the possessive relationship.

Step-by-Step Solution:
Step 1: Identify the intended meaning: work done during a single hard day.
Step 2: Recognize that day is singular in this context, introduced by the article a.
Step 3: Apply the rule for possessive singular nouns: add apostrophe s to day, giving day's.
Step 4: Form the complete phrase: a hard day's work.
Step 5: Compare all options and choose the one that matches this standard phrase.

Verification / Alternative check:
The expression a hard day's work is familiar and widely used in English. We can test it in a new sentence: He felt very tired after a hard day's work. This sentence sounds perfectly natural. On the other hand, after a hard days work looks wrong because no possessive marker is present, and after a hard days' work wrongly uses the plural possessive with a single day. These quick checks confirm that hard day's work is the correct improvement.

Why Other Options Are Wrong:
Option B, hard days' work, uses the plural possessive, which would require a phrase like after several hard days' work. It does not match the article a in the sentence. Option C, hard day work, lacks the apostrophe and does not represent standard idiomatic use. Option D, no improvement, keeps the original error and fails to mark the possessive relationship. Option E, hard working day, changes the meaning and the structure of the noun phrase, and it is not the usual idiom used to talk about the effort spent on one day of work.

Common Pitfalls:
Many learners feel unsure about where to place the apostrophe when expressing time related quantities such as a day's work, a month's salary, or two weeks' notice. Remember that with singular units we use apostrophe s, and with plural units already ending in s we place the apostrophe after the s. Practising these patterns with common time nouns helps avoid errors in exams and in practical communication.

Final Answer:
The correct improvement is hard day's work, so the sentence should read: After a hard day's work, I just want to go home.