We are given that value of x satisfies equation

    x*(x-3) = -1.           (1)


In other words, x is the root of the polynomial  

    p(x) = x^2 - 3x + 1.    (2)


They ask to find the value of polynomial  Q(x) = x^3*(x^3-18) = x^6 - 18x^3.


To do it, make long division and find the REMAINDER of division Q(x) by p(x)

    Q(x) =  - 1.    (3)


Thus, the remainder is -1.


    It means, that if you substitute the root x of the polynomial  p(x) = x^2 -3x + 1 

    into polynomial Q(x), then the first addend in the right side of (3) will be 0 (zero),

    and you will get the final value of -1 for Q(x).


ANSWER.  The value of    is -1.

The post-solution note

Cube both sides:

eq. 1 
eq. 2 

Go back to 


If we multiply both sides by -9 we'll have the two middle terms of the 2nd
factor on the left side of eq. (2)

Let's add the other two terms of the 2nd factor on the left of eq. (2) to both
sides:


Multiply both sides by 


Since the left side = -1 by eq. 1, the right side also = -1.

Edwin