Substitute x = 0 , y = 1 {\displaystyle x=0,y=1} into the equation, we have d = 1 {\displaystyle d=1} .
Substitute x = 1 , y = 2 {\displaystyle x=1,y=2} into the equation, we have a + b + c + 1 = 2. {\displaystyle a+b+c+1=2.}
That is, a + b + c = 1. {\displaystyle a+b+c=1.}
When x = − 1 , y = 4 {\displaystyle x=-1,y=4} , we have − a + b − c + 1 = 4 {\displaystyle -a+b-c+1=4} .
That is, − a + b − c = 3. {\displaystyle -a+b-c=3.}
Solving these equations, we have b = 2 , a + c = − 1 , d = 1 {\displaystyle b=2,a+c=-1,d=1} .
Thus, any cubic polynomials with coefficients satisfying a + c = − 1 , b = 2 , d = 1 {\displaystyle \color {blue}a{+c=-1,b=2,d=1}} are the desired polynomials.
into the equation, we have 
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are the desired polynomials.
