f(x) = x^4 - 97x^2 + 1296
The multiplicity part just means 'how many of each zero is there?'
If the problem was g(x) = (x-2)^2, factoring to g(x) = (x-2)(x-2) there would be just one zero (x=2), with a multiplicity of 2.
x^4 - 97x^2 + 1296 = 0
x^2 = (97 +- sqrt(97^2 - 4*1*1296) ) /2
x^2 = (97 +- sqrt(9409 - 5184) ) /2
x^2 = (97 +- sqrt(4225) )/2
x^2 = (97 +- 65) / 2
x^2 = 162/2 or 32/2
x^2 = 81 or 16
x = -9, 9, -4, or 4
f(x) has zeros of -9, 9, -4, and 4, each with a multiplicity of 1