Can someone explain this math to me

Question

I cant figure out this math how are they doing this and coming up with the numbers they have.

Answer 1

These operations seem pretty straight forward to me.

For the polynomial multiplication $(x^3 + x^2 + 2)(x^2 - x + 1)$ it splits the multiplication into 3 steps.

$(x^3 + x^2 + 2)(x^2 - x + 1) =$

$(x^3 + x^2 + 2)(x^2) + (x^3 + x^2 + 2)(-x) + (x^3 + x^2 + 2)(1)$

As for division, remember that the operation being performed is $\frac{x^3 + x^2 + 2}{x^2-x+1}$ and every step the numerator is being deduced by a factor of the denominator, which appear on the top.

I hope these hints explained how the operations are performed.

Answer 2

These are just the algorithms for integer multiplication and division taught to most elementary schoolkids, with powers of $x$ rather than powers of $10$. In fact that makes them easier, since you never have to carry when adding or borrow when subtracting.

Related: What actually is a polynomial?