What would be a quartic equation whose first and second derivatives factor

Question

Non zeros

Answer 1

y= x^4

y' = 4x^3 factors into 4 factors (4)(x)(x)(x)

y" = 12x^2 factors into (12)(x)(x)

or if you want to work this backwards with a more complicated solution

y" = 12(x+1)(x+1) = 12x^2 + 24x + 12 has 3 factors: 12, x+1, and x+1

y' = 4x^3 +12x^2 + 12x = 4x(x^2 + 3x + 3) has 3 factors: 4, x, and x^2+3x+3

y = x^4 + 4x^3 + 6x^2 + c where c is any constant