Plz answer its related to theory of equation . I cannot understand how value of a is decided

We check the derivative and equate it to 0
we get
x⁴-1 = 0 => maxima at x=-1 and minima at x=1 (check with 2^ndderivative)
so now we can vizulize the curve and if there is only 2 peaks => it will have either one root or three roots based on values at minima and maxima. If they are of the same sign then one root otherwise 3 roots

the question is playing with f(1) = a-4 and f(-1) = a+4
now if opposite signs
f(1).f(-1) <0 => -4 < a < 4 => three real roots