If one factor of polynomial x ^ 4 + MX ^ 2 + 3x + 4 is x ^ 2-x + 4, find the value of M and another factor

If one factor of polynomial x ^ 4 + MX ^ 2 + 3x + 4 is x ^ 2-x + 4, find the value of M and another factor


Let another factor be x ^ 2 + ax + B, then:
(x^2-x+4)(x^2+ax+b)=x^4+ax^3+bx^2-x^3-ax^2-bx+4x^2+4ax+4b
=x^4+(a-1)x^3+(b-a+4)x^2+(4a-b)x+4b
According to the meaning of the question, we get: x ^ 4 + MX ^ 2 + 3x + 4 = x ^ 4 + (A-1) x ^ 3 + (B-A + 4) x ^ 2 + (4a-b) x + 4b, then
a-1=0;
b-a+4=m;
4a-b=3
4b=4
The solution is: a = 1; b = 1; m = 4

Elden Ring Premiere

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