(1) A cubic equation can have up to 3 distinct solutions. If they are in geometric progression, we can write them as u, ur, and ur^2. If those are the solutions to a cubic equation with this equation, you must have (x- u)(x- ur)(x- ur^2)= x^3- 7x^2- 21x+ a. Multiply out on right and compare the coefficients of x^2 and x on left and right to get two equations for u and r. Once you have those it is easy to find a.
(2) Pretty much the same thing. A quartic equation can have up to 4 distinct solutions. If the solutions are in arithmetic progression, we can write them as u, u+ r, u+ 2r, and u+ 3r. So we must have (x- u)(x- u- r)(x- u- 2r)(u+ 3r)= x^4- (a+ 3)r^2+ (a+ 2).