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Factorisable quadratics

The quadratic $x^2+4x+3$ factorises as $(x+1)(x+3)$. In both the original quadratic and the factorised form, all of the coefficients are integers.
The quadratic $x^24x+3=(x1)(x3)$ similarly factorises with all of the coefficients being integers.
How many quadratics of the following forms factorise with integer coefficients? Here, $b$ is allowed to be any integer (positive, negative or zero). For example, in part a, $b$ could be $7$, since $(x2)(x5)=x^27x+10$.
a. $x^2+bx+10$
b. $x^2+bx+30$
c. $x^2+bx8$
d. $x^2+bx16$
e. $2x^2+bx+6$
f. $6x^2+bx20$

This time, it is the constant which is allowed to vary.
How many quadratics of the following forms factorise with integer coefficients? Here, $c$ is allowed to be any positive integer.
a. $x^2+6x+c$
b. $x^210x+c$
c. $3x^2+5x+c$
d. $10x^26x+c$
 What are the answers to question 2 if $c$ is only allowed to be a negative integer?
Generalising
Can you generalise your answers to the above questions?
a. 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.
b. 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.
c. How can we generalise question 2 or 3?
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