We'll first want to rewrite the polynomials as single expressions.
(x3+x+3)(x−1)=x4−x3+x2−x+3x−3
x4−x3+x2−x+3x−3=x4−x3+x2+2x−3
(x−5)2=(x−5)(x−5)=x2−10x+25
Now we need to use long division to find our answer.
Step 1: x2 goes into x4, x2 times, so we need to multiply our divisor, x2−10x+25, by x2, and subtract that from the dividend, x4−x3+x2+2x−3.
x2(x2−10x+25)=x4−10x3+25x2
(x4−x3+x2+2x−3)−(x4−10x3+25x2)=9x3−24x2+2x−3
Step 2: x2 goes into 9x3, 9x times. Repeat step 1 with these values.
9x(x2−10x+25)=9x3−90x2+225x
(9x3−24x2+2x−3)−(9x3−90x2+225x)=66x2−223x−3
Step 3: x2 into 66x2, 66 times. Repeat step 1.
66(x2−10x+25)=66x2−660x+1650
(66x2−223x−3)−(66x2−660x+1650)=437x−1653
Our three divisors are then added together to find our value, x2+9x+66. However, we have remainder of 437x−162, so our answer is x2+9x+66+437x−165x2−10x+25
