In the following question, choose the alternative that best improves the bracketed part of the sentence. In case no improvement is needed, select "no improvement". Rahul went out without (saying) good bye.

Introduction / Context:
This sentence improvement question asks whether the bracketed part "saying" is correct in the structure or whether one of the alternative forms should replace it. The sentence uses the preposition "without" followed by a phrase that describes what Rahul did not do before going out. Understanding how "without" combines with verbs is important for choosing the correct form.

Given Data / Assumptions:

• Sentence: "Rahul went out without (saying) good bye."
• Options: "say even", "even said", "say", "no improvement".
• The meaning is that Rahul left but did not say goodbye.

Concept / Approach:
The preposition "without" is normally followed by a noun or a gerund (verb plus ing) used as a noun. In this sentence, "saying goodbye" functions as an activity that did not occur. The form "without saying goodbye" is a very common and correct pattern in English. The alternatives either disturb the structure or produce ungrammatical combinations, so the original phrase does not require improvement.

Step-by-Step Solution:
Step 1: Examine the original phrase: "without saying good bye". It follows the pattern "without" plus gerund, which is grammatically correct. Step 2: Test option (a) "say even": "without say even good bye" is ungrammatical and has incorrect word order. Step 3: Test option (b) "even said": "without even said good bye" is also ungrammatical because "without" should not be followed directly by a past tense verb here. Step 4: Test option (c) "say": "without say good bye" breaks the rule that "without" requires a noun or gerund and sounds incorrect. Step 5: Since none of the alternative phrases improve the sentence, and the original is grammatically correct, select "no improvement".

Verification / Alternative check:
Compare with standard expressions: "He left without saying a word", "She went to bed without brushing her teeth", and "They left without paying the bill". All of these use "without" followed by a gerund. The given sentence fits exactly the same pattern: "Rahul went out without saying good bye." Thus, common usage confirms that the original form is correct and does not need any change.

Why Other Options Are Wrong:

• Say even: Produces the odd structure "without say even good bye", which is grammatically incorrect.
• Even said: Creates the phrase "without even said good bye", which does not fit standard English syntax.
• Say: "Without say good bye" is ungrammatical because "say" in base form cannot directly follow "without" in this context.
• No improvement: Correctly indicates that "without saying good bye" is already the standard and correct construction.

Common Pitfalls:
Some learners are unsure whether gerund forms are correct after prepositions and may feel tempted to replace them with a simple verb. Others may be drawn to options with the word "even", thinking they make the sentence more expressive. However, examinations focus on grammatical correctness; adding "even" is unnecessary and in the given options is not integrated correctly. The safest strategy is to remember that "without" is normally followed by a gerund when referring to an action that does not take place.

Final Answer:
The sentence is already correct, so the right choice is no improvement.