This can have 0, 2 or 4 real roots, for them to be arithmetic progression there must be 4.

put , then we have:

and for the original equation to have four real roots this must have two real roots. These are:

for these to be real and distinct we require that .

For these to be in arithmetic progression there must be an a such that these roots are

so:

Alternative method:

The roots must be , for some real and so:

Now expand the product on the right, and equate the coefficients of like powers on the left and right to solve for

.