Let us infer from the above words which would seem to imply that the intent of the verse is to command us to erase Amalek from this world, and to destroy and wipe out their memory? If so, the verse should have said 'erase the memory of Amalek, and forget them! Instead our verse says to wipe them out, and do not forget! This seems like a contradiction (to fully wipe them out we must also forget about them!)
And we can say with the help of Heaven, that the intent and mission of Amalek, which is also synonymous with the evil inclination is the following -
כֵּיוָן דְּיִשְׁתָּדַּל בַּר נָשׁ בִּרְעוּתָא דְּלִבָּא לְגַבֵּי קוּדְשָׁא בְּרִיךְ הוּא, לָא בָּעֵי מִנָן אֶלָּא לִבָּא
Since a person has a desire from the heart to be one with HaShem.. so this Yetzer Hara is only here to come after our heart..
The evil inclination tries to separate the children of Israel from their attachment to HaShem through the strategy of entering the attribute of forgetfulness into our hearts, so we forget HaShem and His Torah, G-D forbid. And he comes at people with all forms of trickery, which is hinted to in his name Amalek!
As the numerical value of Amalek is the same as the Hebrew word Ram, meaning elevated, as he tries to inflate a mans heart with his ego, which is the source and foundation for all evil, may HaShem have mercy.
And on this HaShem says -
אָמַר הַקָּדוֹשׁ בָּרוּךְ הוּא אֵין אֲנִי וָהוּא יְכוֹלִין לָדוּר בָּעוֹלָם
The Holy One, Blessed be He, said: He and I cannot dwell together in the world
And similarly Amalek is also the numerical value of the word Mar, bitter, as he tried to enter bitterness and sadness into us. As well, the word Amalek is also the numerical value of the word Safek, doubt, to cause us to have doubts about our faith in HaShem, as its written -
וַיִּקְרָא֙ שֵׁ֣ם הַמָּק֔וֹם מַסָּ֖ה וּמְרִיבָ֑ה עַל־רִ֣יב ׀ בְּנֵ֣י יִשְׂרָאֵ֗ל וְעַ֨ל נַסֹּתָ֤ם אֶת־יְהֹוָה֙ לֵאמֹ֔ר הֲיֵ֧שׁ יְהֹוָ֛ה בְּקִרְבֵּ֖נוּ אִם־אָֽיִן׃ {פ}
As well, Amalek is the numerical value of Atzlan, lazy, as he tries to bring laziness into a persons service of HaShem. And all of this is with the intention of of bringing the people of Israel to a place where they forget about HaShem and His Torah, G-D forbid. This is why our Holy Torah commands us to erase the memory of Amalek from under the heavens, which means to say that to erase this evil inclination, which in essence is all these negative attributes we host within us that are all hinted to within the numerical value of Amalek... and then you will be worthy to the end of the verse 'Don't forget', meaning that you wont come to a place of forgetfulness of HaShem and His Holy Torah, and you will attach yourself to eternal life, and this is correct with the help of Heaven.