When you divide by a fraction, you can flip the denominator of the whole thing and multiply:
4a/(2a^3 + 2) * (4a + 4)/8a
(16a^2 + 16a)/(16a^4 + 16a)
You can factor out a 16a/16a:
(a + 1)/(a^3 + 1)
a^3 + 1 is a sum of cubes; it's equal to (a + 1)(a^2 - a + 1). (a + 1) cancels from that and the numerator, and you should be left with 1/(a^2 - a + 1).
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Where is your equal sign?
When you divide by a fraction, you can flip the denominator of the whole thing and multiply:
4a/(2a^3 + 2) * (4a + 4)/8a
(16a^2 + 16a)/(16a^4 + 16a)
You can factor out a 16a/16a:
(a + 1)/(a^3 + 1)
a^3 + 1 is a sum of cubes; it's equal to (a + 1)(a^2 - a + 1). (a + 1) cancels from that and the numerator, and you should be left with 1/(a^2 - a + 1).