Μέχρι το τέλος του μήνα η βασική γραμματική θα έχει εξηγηθεί πολλές φορές στο μάθημα.

Θα έχει εξηγηθεί is future perfect passive.

• Tense: future perfect → “will have been explained”
• Voice: passive → the subject (η βασική γραμματική) receives the action.

Formation pattern (passive voice):

• Present passive: εξηγείται – is explained / is being explained
• Perfect passive: έχει εξηγηθεί – has been explained
• Future perfect passive: θα έχει εξηγηθεί – will have been explained

Structure:

• θα (future particle)
• • έχει (3rd person singular of έχω)
• • εξηγηθεί (aorist passive form of εξηγώ)

So θα έχει εξηγηθεί literally = “it will have been explained.”

Both are possible, but they don’t say exactly the same thing.

• Θα εξηγηθεί πολλές φορές
  → “It will be explained many times.”
  Focus: the explanation will happen in the future (repeatedly), without strongly stressing that it will already be complete by a certain deadline.

• Θα έχει εξηγηθεί πολλές φορές
  → “It will have been explained many times.”
  Focus: by the time we reach the end of the month, this process will already be completed (it will already have happened several times).

Because the sentence starts with Μέχρι το τέλος του μήνα (By / Until the end of the month), Greek very naturally uses the future perfect to highlight that at that future point, the repeated explanations will be a finished fact.

Μέχρι το τέλος του μήνα means “by the end of the month / until the end of the month” and sets a time limit.

Breakdown:

• μέχρι = until / up to / by
• το τέλος = the end (neuter, accusative singular)
• του μήνα = of the month (masculine, genitive singular)

So grammatically:

• μέχρι is followed by the accusative (το τέλος).
• του μήνα is genitive, showing possession/association: “the end *of the month”*.

Meaning-wise, μέχρι here introduces a deadline in the future, which fits perfectly with the future perfect: by that point in time, the action (explaining the basic grammar) will already be completed several times.

In English, “basic grammar” is the object of “explain”.
In passive Greek (and English), the object of the active verb becomes the subject.

Active version (for comparison):

• Ο καθηγητής εξηγεί τη βασική γραμματική.
  – The teacher explains the basic grammar.
  
  • Ο καθηγητής = subject (nominative)
  • τη βασική γραμματική = direct object (accusative)

Passive:

• Η βασική γραμματική εξηγείται.
  – The basic grammar is explained.
  Now:
  • η βασική γραμματική becomes the subject of the passive verb, so it is in nominative.

In our sentence:

• η βασική γραμματική θα έχει εξηγηθεί πολλές φορές
  → η βασική γραμματική is the subject of the passive verb θα έχει εξηγηθεί, so it must be nominative feminine singular.

Εξηγηθεί is a verbal form used with έχω to form perfect tenses in the passive.

For εξηγώ (to explain), passive forms look like this:

• Aorist passive: εξηγήθηκα – I was explained / I got an explanation (roughly)
• Perfect passive: έχω εξηγηθεί – I have been explained / It has been explained

Here, εξηγηθεί is the aorist passive stem that combines with έχω (or θα έχει) in periphrastic tenses:

• έχει εξηγηθεί – has been explained
• θα έχει εξηγηθεί – will have been explained

Εξηγημένη is a participle/adjective (explained as an adjective), and you use it differently:

• Η γραμματική είναι εξηγημένη. – The grammar is explained (adjectival result, like finished).

In our sentence we need a verb phrase (will have been explained), not just an adjective, so εξηγηθεί is the correct form.

Because here “many” modifies a countable noun (φορές – times):

• φορές = times (feminine plural noun)
• πολλές = many (feminine plural adjective) agreeing with φορές

So:

• πολλές φορές = many times

Πολύ can be:

• an adverb: πολύ καλά – very well
• or an invariable adverb meaning “a lot”: δουλεύει πολύ – he/she works a lot

You can’t use πολύ as an adjective before φορές; you need the adjective πολλές in the correct gender and number:

• ✔ πολλές φορές
• ✘ πολύ φορές

Του μήνα is genitive singular of ο μήνας (the month).

• Nominative: ο μήνας
• Genitive: του μήνα

In το τέλος του μήνα, του μήνα is in the genitive because Greek uses the genitive to express:

• possession: the book of the child
• relationships like “the end of X”, “the beginning of X”, etc.

So το τέλος του μήνα literally = “the end of the month”, with του μήνα depending on το τέλος.

Στο is a contraction of:

• σε (preposition: in, at, to)
• • το (neuter definite article: the)

So:

• σε + το = στο

Στο μάθημα literally = “in the lesson / at the lesson”, and in natural English: “in class / during the lesson.”

Other similar contractions:

• σε + τον = στον (masculine)
• σε + την = στην (feminine, usually written στη(ν))
• σε + τους = στους, σε + τις = στις, etc.

Both can sometimes be translated as “in class”, but they focus on different ideas.

• στο μάθημα

  • Literally: in the lesson
  • Refers to the lesson as an event / teaching session / subject.
  • In our sentence: πολλές φορές στο μάθημα = many times during the class/lesson.

• στην τάξη

  • Literally: in the classroom or in the class (group of students)
  • Focuses more on the physical place or the class group, not specifically the teaching content.

Here, because we mean during the lessons, στο μάθημα is the more natural choice.

Modern Greek word order is fairly flexible, though neutral patterns are preferred.

Original:

• Μέχρι το τέλος του μήνα η βασική γραμματική θα έχει εξηγηθεί πολλές φορές στο μάθημα.

Other acceptable orders (with slightly different emphasis):

1. Η βασική γραμματική θα έχει εξηγηθεί πολλές φορές στο μάθημα μέχρι το τέλος του μήνα.
   – Puts the subject first; the time limit moves to the end.

2. Η βασική γραμματική, μέχρι το τέλος του μήνα, θα έχει εξηγηθεί πολλές φορές στο μάθημα.
   – Commas highlight μέχρι το τέλος του μήνα as a parenthetical time frame.

3. Πολλές φορές στο μάθημα θα έχει εξηγηθεί η βασική γραμματική μέχρι το τέλος του μήνα.
   – Emphasizes how often / where first.

All of these remain grammatical. Greek often likes putting time expressions at the beginning (as in the original) to set the time frame.

A natural active equivalent would be:

• Μέχρι το τέλος του μήνα θα έχουμε εξηγήσει τη βασική γραμματική πολλές φορές στο μάθημα.

Breakdown:

• θα έχουμε εξηγήσει = future perfect, active voice: we will have explained
• τη βασική γραμματική = direct object (accusative)
• Subject (we) is implied in θα έχουμε.

Meaning: “By the end of the month, we will have explained the basic grammar many times in class.”

So:

• Passive: η βασική γραμματική θα έχει εξηγηθεί – the basic grammar will have been explained
• Active: θα έχουμε εξηγήσει τη βασική γραμματική – we will have explained the basic grammar