Question 4: Generalising question 1, if c is a fixed integer, how many quadratics of the form x^2 + bx + c factorise with integer coefficients? Here, b is allowed to be any integer.
Answer: The number of b = the number of factors of c
Question 5: Further generalising question 1, if a and c are fixed integers, with a positive, how many quadratics of the form ax^2 +bx + c factorise with integer coefficients? Again, b is allowed to be any integer.
Answer: The number of b = the number of factors of (a multiplied by c)
Question 6: How can we generalise question 2 or 3?
Answer:
In question 2, the number of c = the number of pairs of factors that add up to b, a cannot be higher than b.
The answer to question 3 is always infinity, and therefore any quadratics with the form ax^2 +bx + c, with c varying and only allowed to be a negative integer, the number of c = infinity.