Question

If the rest of the division of f ( x ) by x³ - 3x + 5 is 3x²-2 and
the remainder of the division ( x² + f ( x ) ) ^ 2 by x³ - 3x + 5 is ax ² + bx + c ,
then a + b + c = ...

Answer 1

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answer is attached to the file

thanks

Answer 2

The answer is  the option (b) - 44...
This is certainly Degree or PG level..
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f(x) = unknown polynomial.   Let g(x) = x³ - 3 x + 5

Given  f  Modulo g  = 3 x² - 2

[x² + f ]² Modulo  g
    = { x⁴ + 2 x² * [f modulo g ] +  [ (f modulo g)²  modulo g ]  } modulo g
    = { x⁴ + 2 x² (3x² - 2) + [ (3x² - 2)² modulo g]  } modulo g
    = { 7 x⁴ - 4 x² + [ (9x⁴ - 12 x² + 4) modulo g]  } modulo g
    = { 16 x⁴ - 16 x² + 4 } modulo g
    = 32 x² - 80 x + 4
    = a x² + b x + c    .... given

compare coefficients  :  a = 32,   b = - 80    c = 4
=> a + b + c = - 44.