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?
Remember that velocity is a vector and vectors have a magnitude and a direction. The direction of v is the unit vector in the direction of 3i - 4j. The magnitude of v is its speed 10 m/s. So you multiply the unit vector by 10 (the magnitude of the velocity) to get the velocity vector.
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.