Çözüldü Polynomial Division - Undetermined Coefficients Rule - 4 Variable Linear Equation

A question from a Facebook group https://www.facebook.com/groups/1174585619345646;

If a cubic polynomial gives remainders 5x - 7 and 12x - 1 when divided by x² - x + 2 and x² + x - 1 respectively, find the polynomial.

Let the polynomial be ax^3 + bx^2 + cx + d,
As homework to the interested students;
when divided by x^2 - x + 2 the remainder is (-a + b + c)x - 2a - 2b + d = 5x - 7,
upon division by x^2 + x - 1 the remainder is (2a - b + c)x - a + b + d = 12x - 1,
Then applying Undetermined Coefficients Rule;
-a + b + c = 5
-2a - 2b + d = -7
2a - b + c = 12
-a + b + d = -1
Solving this four-variable linear equation (again homework for the studious pupils), the coefficients are;
a = 3, b = 1, c = 7, d = 1
So the cubic polynomial is 3x^3 + x^2 + 7x + 1.

Checks by WolframAlpha:
https://www.wolframalpha.com/input?i=remainder of (3x^3+x^2+7x+1)/(x^2-x+2)
https://www.wolframalpha.com/input?i=remainder of (3x^3+x^2+7x+1)/(x^2+x-1)