Нет бы проверить бланк ответов — мы ошиблись.

In this expression, нет бы is an idiomatic phrase, not the ordinary literal нет meaning no.

The pattern нет бы + infinitive expresses something like:

• you/I/we should have just...
• if only you/I/we had...
• instead of that, it would have made sense to...

It usually carries a tone of regret, annoyance, or mild reproach.

So Нет бы проверить бланк ответов means roughly:

• We should have checked the answer sheet
• If only we had checked the answer sheet
• Wouldn’t it have been smart to check the answer sheet

It often implies that the sensible action was obvious, but it was not done.

Because the construction is нет бы + infinitive.

So after нет бы, Russian commonly uses the infinitive to name the action that would have been the smart or desirable one.

Examples of the same pattern:

• Нет бы спросить. — You/I should have asked.
• Нет бы подождать. — You/I should have waited.
• Нет бы проверить бланк ответов. — We should have checked the answer sheet.

This infinitive does not mean the action is happening now. In this construction, it refers to an action that would have been better to do.

The subject is often left unstated with нет бы + infinitive, because it can be understood from context.

In your sentence:

• Нет бы проверить бланк ответов — мы ошиблись.

the second part gives the subject explicitly: мы.

So the implied meaning is:

• Нет бы нам проверить бланк ответов...
• We should have checked the answer sheet...

Russian often leaves out words that are easy to infer. Here, the speaker assumes the listener can connect the first clause with мы from the second clause.

This is about aspect.

• проверить = perfective
• проверять = imperfective

Here, проверить is used because the speaker means do the check as a complete action. The idea is not be checking or check repeatedly, but carry out the check once and get it done.

That fits the situation:

• Нет бы проверить бланк ответов...
  = We should have checked the answer sheet (once, properly, as a completed step).

If you used проверять, it would sound less natural here, because the point is the failure to perform one concrete, completed action.

Бланк ответов literally means answer sheet.

Breakdown:

• бланк = form / sheet
• ответов = genitive plural of ответы (answers)

So бланк ответов is a very common Russian noun-noun pattern where the second noun goes into the genitive to show what kind of form/sheet it is:

• бланк ответов = sheet of answers / answer sheet
• compare:
  • стакан воды = a glass of water
  • карта города = a map of the city

Here ответов is genitive plural because the sheet contains or is intended for answers in general.

Also note that бланк here is the direct object of проверить, so it stays in the accusative, but since it is an inanimate masculine noun, its accusative form is the same as the nominative: бланк.

Мы ошиблись means we made a mistake or we were mistaken.

It comes from the verb ошибиться, which is a very common way to express making a mistake.

This is more natural than мы сделали ошибку in many contexts.

Compare:

• мы ошиблись = natural, common, idiomatic
• мы сделали ошибку = understandable, but often less idiomatic in ordinary speech

Russian prefers the verb ошибиться in many situations where English uses make a mistake.

The ending -сь is because the verb is reflexive in form: ошибиться.

The dash links the two ideas in a vivid, conversational way.

• Нет бы проверить бланк ответов — мы ошиблись.

The relationship is something like:

• We should have checked the answer sheet — and as a result, we made a mistake.
• If only we had checked the answer sheet; instead, we made a mistake.

The dash helps show a jump from the missed sensible action to the bad outcome.

It gives the sentence a spoken, expressive feel. A comma would usually be weaker here. The dash makes the contrast/result more dramatic.

It is mostly conversational and somewhat emotional.

The construction нет бы + infinitive is common in speech when someone is:

• criticizing themselves or others
• showing frustration
• expressing hindsight

So this is not especially formal. In a formal explanation, you would more likely say something like:

• Нам следовало проверить бланк ответов.
• Надо было проверить бланк ответов.

These are more neutral ways to say we should have checked the answer sheet.

By contrast, Нет бы проверить бланк ответов... sounds more like:

• And of course, instead of checking the answer sheet...
• We really should have checked the answer sheet...

This is an important nuance question.

Надо было проверить

This is the more neutral statement:

• We should have checked.

It can simply state obligation in hindsight.

Нет бы проверить

This adds emotion:

• regret
• annoyance
• self-criticism
• reproach

It often suggests:

• that was the obvious thing to do
• but we didn’t do it
• and now look what happened

So:

• Надо было проверить бланк ответов.
  = neutral hindsight

• Нет бы проверить бланк ответов.
  = emotionally colored, almost like Why didn’t we just check the answer sheet?

Yes. It can be used for:

• yourself
• another person
• a group including yourself
• sometimes even in a more general sense

Examples:

• Нет бы мне раньше выйти.
  I should have left earlier.

• Нет бы тебе позвонить.
  You should have called.

• Нет бы нам проверить всё заранее.
  We should have checked everything in advance.

In your sentence, it is about мы: the speaker includes themselves in the criticism.

Not really, because a literal word-for-word translation would miss the idiom.

Even though нет by itself usually means no / there is not, the combination нет бы functions as a set expression with its own meaning.

So you should understand нет бы here as a whole, not by translating each word separately.

That is very common in Russian: some small combinations develop idiomatic meanings that are much more important than the literal meanings of the individual words.

So the best approach is:

• learn нет бы + infinitive as a fixed pattern
• associate it with should have / if only / instead of that, one should have...

Yes, that is possible, and it can make the subject more explicit.

• Нет бы нам проверить бланк ответов.

This means the same basic thing, but with нам the sentence spells out who should have done it.

Still, Russian often omits that pronoun/dative phrase if the context already makes it clear. In your original sentence, мы ошиблись immediately tells you who is involved, so leaving out нам sounds natural.

So:

• Нет бы проверить бланк ответов — мы ошиблись.
  natural, concise, conversational

• Нет бы нам проверить бланк ответов — мы ошиблись.
  also correct, slightly more explicit