Unit vector in direction of 3i - 4j is (3i - 4j)/5. Therefore v = 10 (3i - 4j)/5 = 6i - 8j.
Therefore dx/dt = 6i - 8j => x = 6ti - 8tj + C.
When t = 0, x = 2i + j therefore C = 2i + j.
Therefore x = (6t + 2) i + (1 - 8t) j.
I understand the rest, bar how you managed to come about the two statements highlighted in red. Why is the first divided by and, and the second doubled?
How to get a unit vector: Divide the given vector by its magnitude. The magnitude of 3i - 4j is 5. Therefore you divide 3i - 4j by 5 to get a unit vector in the direction of 3i - 4j.
10 (3i - 4j)/5 = 2 (3i - 4j) = 6i - 8j.