Another approach is to look at the polynomial you seek to solve / factorise and separate the terms that can take on a odd value (restricting yourself to integers). Looking at the polynomial we have, every term will be even when x is an even integer, but when x is odd, there are only two terms that have an odd value, x4 and 19x2. So, we can say:
I can see that any odd integer must give a value of that is a multiple of 6, which allows for a quick search for odd integer solutions.
Case:
So, neither nor is a solution.
Case:
So, we have one solution, .
Case:
So, we have a second solution, .
Having found two factors, we can divide by to find the second quadratic factor, which will be , but other approaches are faster.
We know that we can take the rots to be , , , and . We know two of these, so set and . We also know (from the constant being positive) that there are an even number of positive roots, and so one of the two roots we are yet to find is positive and the other is negative.
It immediately follows that the quadratic with roots and is which is also what the long division would find.
Thus, we have:
and thus the solutions of are .
I can see that any odd integer must give a value of that is a multiple of 6, which allows for a quick search for odd integer solutions.
Case:
So, neither nor is a solution.
Case:
So, we have one solution, .
Case:
So, we have a second solution, .
Having found two factors, we can divide by to find the second quadratic factor, which will be , but other approaches are faster.
We know that we can take the rots to be , , , and . We know two of these, so set and . We also know (from the constant being positive) that there are an even number of positive roots, and so one of the two roots we are yet to find is positive and the other is negative.
It immediately follows that the quadratic with roots and is which is also what the long division would find.
Thus, we have:
and thus the solutions of are .