If a, b, p, q are all rational then there can be no solution. Here's why:
Because this is an equation, the left and right sides must balance. If we take the coefficients of the highest degree of x (x20) we have the equation 220-a20=1, so a20=220-1. a would therefore be irrational. a would also be very close to 2 (about 1.999999905).
Similarly if we take the constant term on both sides we get the equation 1-b20=q10, 0≤b20≤1, since q10 has to be positive (raised to an even power). If b=1, q=0 and if b=0, q=1. Other fractional values of b would make q an irrational fraction. For example, if b=0.9, q=0.98712 approx.