We'll first want to rewrite the polynomials as single expressions.
(x^3 + x + 3)(x-1) = x^4 - x^3 +x^2 - x +3x - 3
x^4 - x^3 +x^2 - x +3x - 3 = x^4 - x^3 +x^2 + 2x - 3
(x-5)^2 = (x - 5)(x - 5) = x^2 - 10x + 25
Now we need to use long division to find our answer.
Step 1: x^2 goes into x^4, x^2 times, so we need to multiply our divisor, x^2 - 10x + 25, by x^2, and subtract that from the dividend, x^4 - x^3 +x^2 + 2x - 3.
x^2 (x^2 - 10x + 25) = x^4 - 10x^3 +25x^2
(x^4 - x^3 +x^2 + 2x - 3) - (x^4 - 10x^3 +25x^2) = 9x^3 -24x^2 + 2x - 3
Step 2: x^2 goes into 9x^3, 9x times. Repeat step 1 with these values.
9x(x^2 - 10x + 25) = 9x^3 - 90x^2 + 225x
(9x^3 -24x^2 + 2x - 3) - (9x^3 - 90x^2 + 225x) = 66x^2 - 223x - 3
Step 3: x^2 into 66x^2, 66 times. Repeat step 1.
66(x^2 - 10x + 25) = 66x^2 - 660x + 1650
(66x^2 - 223x - 3) - (66x^2 - 660x + 1650) = 437x - 1653
Our three divisors are then added together to find our value, x^2 + 9x + 66. However, we have remainder of 437x - 162, so our answer is x^2 + 9x + 66 + (437x - 165)/(x^2 - 10x + 25)
