Everybody knows mantharA was the one who poisoned kaikeyi's mind and made her send rAma to exile. But what was mantharA's motivation and why did she have such enmity to rAma?

This is explained by azhwars as a parama-rahasya and is only hinted in smR^iti. Will produce both here+

First, the words of thirumazhisai azhwar.

kUnagaM pugazh therindha  . koRRa villi allaiyE

[Are you not the master of the victorious bow, which straightened the hump of mantharA and thus by this playful act, was the cause of protection of the worlds.]+

Thirumazhisai azhwar explains the reason for mantharA's animosity. When rAma was a child, he used to have a toy bow & shot mudballs with it.

Whenever he saw mantharA's hump, he shot mud balls at her hump. Her hump got straightened by the shots, then wobbled back into place+

As bhagavAn was a child, he used to do that and laugh at the hump. Since he was a prince, mantharA could not speak against him and so tolerated it. But she harbored a grudge against bhagavAn due to that act.

It led to her taking her chance to turn kaikeyi against rAma+

Thirumazhisai Azhwar also says that toy bow itself was most victorious -- it became the cause of protection, as it led to bhagavAn going to the forest and killing rAvaNa. Thus, his nature shone through even in this instance+

Now, Acharyas say this is a parama-rahasya and was directly seen by Azhwar, hence, it is not even in the smR^itis. Indeed, vAlmiki himself does not give any backstory to mantharA.

But, this incident is hinted at by mArkandeya in MB, vana parva. Quoting his narration+

तेषां समक्षं गन्धर्वी दुन्दुभीं नाम नामतः । शशास वरदो देवो गच्छ कार्यार्थसिद्धये

[In the presence of gods, brahmA, who is "varada", spoke to the gandharvI dundubhI, “Go accomplish the task (of sending rAma to the forest)"]+

brahmA is called "varada" as he gave boons of "rAma-kainkarya" - service to rAma- to the gods. He commanded them to take birth as vAnarAs.

He also commanded dundubhi, "make rAma go into exile". That is the implication of the command as seen below+

पितमहवचः श्रुत्वा गन्धर्वी दुन्दुभी ततःमन्थरा मानुषे लोके कुब्जा समभवत्तदा

[Hearing the words of brahmA, the pitAmaha, Dundubhi was born in the world of humans as a hunchback named mantharA.]+

Note the reference to "kubjA". This establishes what Azhwars say+

brahmA only commanded dundubhi to contrive a means to send bhagavAn to exile. Why did she then take birth as a hunchback to accomplish it?

Implication is -- her hump would become a reason for sending bhagavAn to the forest, as the azhwars say. Hence, mArkandeya mentions "kubjA"+

एवं विधाय तत्सर्वं भगवाँल्लोकभावनः । मन्थरां बोधयामास यद्यत्कार्यं त्वया तथा ।

[Having arranged all this, venerable brahmA, who nourishes the worlds by enabling birth of rAma, instructed mantharA on what had to be done by her (to incur a grudge towards rAma)]+

What did brahmA instruct her to do? mArkandeya does not mention it explicitly as it is a rahasya. But it is this- it is not enough to just be born as a hunchback.

To make a child (bhagavAn) shoot mudballs at it, she has to grab his attention to her hump+

So, brahmA was instructing her to show her hump prominently to child rAma whenever she saw him, to kindle a desire in him to play with it. Only when a child's interest is piqued in a "toy", does the child endeavor to play with the toy.

This was the instruction of brahmA+

सा तद्वच समाज्ञाय तथा चक्रे मनोजवा । इतश्चेतश्च गच्छन्ती वैरसंधुक्षणे रता

[By quickness of mind, mantharA understood everything brahmA told her. And she went about being engaged in inciting enmity (in Kaikeyi towards rAma).]+

"manojavaH" also signifies this. It means, mantharA, being quick of mind, understood how brahmA wanted her to gain a grudge against rAma.

The consequence of that act is mentioned next - she went about creating enmity in kaikeyi towards rAma, which was the completion of her duty+

Now, why is this a parama-rahasya? Because bhagavAn says, "I consider myself rAma, son of dasharatha". He does not like to be identified as paramAtma in that avatAra. So, the gods decided not to reveal that him going to the forest was verily his own sankalpa from the start!+

Finally, Nammazhwar remarks on this incident as well:

kUnE sidhaiya uNdai vil niRaththil theRiththAy gOvindhA

[O Govinda, who straightened the hump of the hunchback by hitting it with mudballs from the bow.]

Note that azhwar refers to rAma as "Govinda"+

Nampillai explains, "normally, when a mistake is committed, the one most well-known for mistakes is accused even if there is no proof. Krishna is most well known for mischief, rAma is not --- so Azhwar laid the blame on Krishna"+

Finally, shrI velukkudi swami observes, "Krishna committed so many acts of mischief and got felicitated for it. For one act of mischief, rAma got everlasting hatred from mantharA and got sent to the forest. The lesson -- one should always stick to things one is competent in"+

"Krishna is competent at mischief. rAma is competent at speaking Truth. Thus, each should abide in their own dharma"

Azhwars had the vision and so revealed what is only hinted at in shAstra//