Express the polynomial in the form p(x) = (x+1) Q(x) +R where (x+1) is the divisor, Q(x) is the quotient and R is the remainder Express the polynomial in the form p(x) = (x+1) Q(x) +R where (x+1) is the divisor, Q(x) is the quotient and R is the remainder, 
Hey I would just like to know how to solve this as the question had me confused I don’t know if I am suppose to do long division on write it out like in the question, thanks 
 A: Any textbook would tell you to do polynomial division. Personally, I think polynomial division is too opaque of a technique to use before you know how to solve a question like this without it. So here is an example using what basically amounts to long division, but much less mysterious.
Take $p(x) = x^3+2x^2-3x+5$. We want to write this of the form $(x+1)Q(x) + R$ for some polynomial $Q$, and some polynomial $R$ with smaller degree than $x+1$ (which is to say, $R$ is a constant). We do this by building $Q$ term by term, reducing the degree of the remainder until we have our constant $R$.
The fact that the highest degree term of $p$ is $x^3$ means that the highest degree term of $Q$ must be $x^2$. So we add $(x+1)x^2$, and simultaneously subtract $x^3+x^2$ from the remainder. This doesn't change the value, but it lets us start on building our $Q$:
$$
p(x) = (x+1)x^2 - (x^3+x^2) + x^3+2x^2-3x+5\\
= (x+1)x^2 + x^2-3x+5
$$
We have now successfully gone from a third degree remainder (all of $p$) to a second degree remainder. We are one step closer to a constant remainder. Next step: The highest degree term of the remainder is $x^2$, so the next term to add to $Q$ must be $x$:
$$
p(x) = (x+1)x^2 - (x^2 + x) + (x+1)x + x^2-3x+5\\
= (x+1)(x^2+x) -4x+5
$$
And finally, in the last step, we see that the leading term of the current remainder is $-4x$, so the final term to add to $Q$ must be $-4$:
$$
p(x) = (x+1)(x^2+x) + (x+1)(-4) - (-4x-4) -4x+5\\
= (x+1)(x^2+x-4) + 9
$$
And there we have it: $Q(x) =  x^2+x-4$ and $R = 9$. If you do the long division, you might recognize how each step in the long division corresponds to something I did here, and vice versa, but long division is something of a mystery if one doesn't know that this is the process that lies behind it.
Also, I have been extra verbose here. If I were to give this problem on a test, I wouldn't necessarily expect more from an answer than, say,

$$
p(x) = x^3+2x^2-3x+5\\
= (x+1)x^2 + x^2-3x+5\\
= (x+1)(x^2+x) -4x+5\\
= (x+1)(x^2+x-4) + 9
$$

All-in-all more transparent and easier to follow than polynomial long division.
A: Put $\,a = -1\,$ in $\ p(x)\, =\, (x-a)\,\dfrac{p(x)-p(a)}{\ x\,-\,a} + p(a),\,$ the Polynomial Remainder Theorem.
The quotient is exact by the Polynomial Factor Theorem, and is computable by the Polynomial Division Algorithm (which also computes the remainder).
