Past Negative

To buy

1. かうませんです
2. かいます
3. かいませんでした
4. かうました

3. かいませんでした

Explanation: In Japanese, when it comes to verbs, you can have them in the informal, or formal form. There are two types of present-affirmative informal verbs. There are "る" verbs (Verb Ends With a "る" Sound), and the "う" verbs (Verb Ends With a "う" Sound). (There are some exceptions). Whenever you have a "る" verb, and you want to make it formal. all you do is get rid of the "る," and add "ます" at the end for the formality.

Examples:

たべる > たべます / To Eat

ねる > ねます / To Sleep

Whenever you have an "う" verb, what you have to do is replace the "う" sound of the verb to an "い" sound. Then you have to add the ます.

Examples:

やすむ > やすみます / To Rest

あそぶ > あそびます / To Play

いう > いいます / To Say

With changing the formality of the verb out of the way, it now comes to changes the past-tense.

In your possible answers, the definitions would be these -

1. かうませんです. This makes no sense at all. First of all, the verb is still in it's dictionary form (negative present-affirmative), with the polite "ます" afterwards. Also, "です" makes no sense to be at the end. It is wrong.

2. かいます. This verb conjugation is the present affirmative, "To Buy." It makes sense, but not correct.

3. かいませんでした. It makes sense, and is in the correcr past-negative form.

4. Again, the verb isn't conjugated. Adding the "ます" doesn't make sense.

Note that some verbs are exceptions. Hope this helps!