divide a polynomial p(x) by (x-3). Add and subtract the multiple of (x-3) that has the same highest-power term as p(x), then simplify to get a smaller-degree polynomial r(x) plus multiple of (x-3).
The multiple of (x-3) that has x^4 as its leading term is x^3(x-3) = x^4 - 3x^3. So write:
x^4 + 7 = x^4 + 7 + x^3(x - 3) - x^3(x - 3)
= x^4 + 7 + x^3(x - 3) - x^4 + 3x^3
= x^3(x - 3) + 3x^3 + 7
That makes r(x) = 3x^3 + 7. Do the same thing to reduce r(x) by adding/subtracting 3x^2(x - 3) = 3x^3 - 9x^2:
= x^3(x - 3) + 3x^3 + 7 + 3x^2(x - 3) - (3x^3 - 9x^2)
= x^3(x - 3) + 3x^2(x - 3) + 9x^2 + 7
Again to reduce 9x^2 + 7:
= x^3(x - 3) + 3x^2(x - 3) + 9x^2 + 7 + 9x(x - 3) - (9x^2 - 27x)
= x^3(x - 3) + 3x^2(x - 3) + 9x(x - 3) + 27x + 7
And finally write 27x + 7 as 27(x - 3) + 88;
x^4 + 7 = x^3(x - 3) + 3x^2(x - 3) + 9x(x - 3) + 27(x - 3) + 88
Factor out (x - 3) in all but the +88 term:
x^4 + 7 = (x - 3)(x^3 + 3x^2 + 9x + 27) + 88
That means that:
(x^4 + 7) / (x - 3) = x^3 + 3x^2 + 9x + 27 with a remainder of 88