Polynomial Division: where does the remainder go? If
 P(x) = x 4 + 4x 3 - 14x 2 + 36x + 45
is divided by x + 5
and the remainder that I found is -1250, would it go at the end of the quotient? Like this:
x 3 - 9x 2 - 59x + 259 -  1250  ?

*...forgot how to do this stuff :\
 A: Since $P(-5)=-360$ and since$$\frac{P(x)+360}{x+5}=x^3-x^2-9x+81,$$you have $P(x)=(x^3-x^2-9x+81)(x+5)+360$.
A: $\,P(x)=q(x).g(x)+r(x)$
:.
    $(x^4+4x^3-14x^2+36x+45) = (x^3-x^2-9x+81).(x+5) + (-360)$.
I hope you asked for this, otherwise pardon.
Alternatively use remainder theorem...
to find remainder only.
$P(-5)=-360.$
A: I suggest you do not trust your memory, think of division from first principles. How does integer division work? If you try to divide $\frac{7}{5}$ you know that the quotient is $1$, but how do you find the remainder? You multiply $1\cdot 5=5$ but you still need $2$ to reach $7$. In other words:
$$ 7 = 1 \cdot 5 + 2$$
or, in general, if you divide $\frac{N}{D}$ obtaining the quotient $Q$ and the remainder $R$, it is true than:
$$ N = Q \cdot D + R$$
Now, if you divide both sides of this equation by $D$ you recover the original fraction:
$$\frac{N}{D}=Q+\frac{R}{D}$$
It works the same with polynomials, so in your case:
$$\frac{P(x)}{x+5}=Q(x)+\frac{R}{x+5}$$
where $R=-360$ as other answers have pointed out.
A: HINT: the right result should be $${x}^{3}-{x}^{2}-9\,x+81-360\, \left( x+5 \right) ^{-1}$$the remainder is given by $-360$
