Question

How might a second-born son hope to achieve success under the system of primogeniture?

He could hope to inherit property from his father.

He could seek training to become a knight in the military.

He could try to attend one of the many trade schools.

He could work on developing his share of his father's land.

Answer 1

How might a second-born son hope to achieve success under the system of primogeniture?
B) He could seek training to become a knight in the military.

Read this for more info:
https://en.wikipedia.org/wiki/Primogeniture

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Primogeniture is the right, by law or custom, of the firstborn son to inherit the family estate, in preference to siblings (compare to ultimogeniture). In the absence of children, inheritance passed to collateral relatives, usually males, in order of seniority of their lines of descent. The eligible descendants of deceased elder siblings take precedence over living younger siblings, such that inheritance is settled in the manner of a depth-first search.

The principle has applied in history to inheritance of real property (land) as well as inherited titles and offices, most notably monarchies, continuing until modified or abolished.

Variations on primogeniture modify the right of the firstborn son to the entirety of a family's inheritance (see appanage) or, in the West since World War II with the wider promotion of feminism, eliminate the preference for males over females. Most monarchies in Europe have eliminated male preference in succession: Belgium, Denmark, Luxembourg, Netherlands, Norway and Sweden. The United Kingdom passed legislation to establish gender-blind succession in 2013 but delayed implementation until the 15 other countries which share the same monarch effect similar changes in their succession laws."

That pretty much gives it away.

Answer 2

He could seek training to become a knight in the military.