For that " $n$ "

We can write the given expression as

$\dfrac {n^3+10}{n+10}=n^2-10n+100-\dfrac {990}{n+10}$

Since the expression is an integer, the maximum possible value of $n$ is $990-10$ or $\boxed {980}$ .

$\begin{aligned} \frac {n^3+10}{n+10} & = \frac {n^2(n+10) - 10n^2+10}{n+10} \\ & = \frac {n^2(n+10) - 10n(n+10) + 100n +10}{n+10} \\ & = \frac {n^2(n+10) - 10n(n+10) + 100(n +10) -990}{n+10} \\ & = n^2 - 10n + 100 - \frac {990}{n+10} \end{aligned}$

For the left-hand side $\dfrac {n^3+10}{n+3}$ to be an integer, $990$ must be divisible by $n+10$ and the biggest $n$ that can do that is $\boxed{980}$ .

To solve this , you have to know one think ...that if A and B both are divisible by P , then both A+B and A-B are also divisible by P . Here I am doing one think if I write k | m then i am meaning that m is divisible by k.
Here,
n+10 | n³+10 ------(1)
Now it is obvious that n+10 | n²(n+10) ----(2) now using (1) and (2) , we can say that n+10 | n²(n+10) -(n³+10)
That means n+10 | 10n²-10 -----(3)..now it is also obvious that n+10 | 10n(n+10) ----(4) now using (3) and (4) we can say that n+10 | 10n(n+10) -(10n²-10) ..that means n+10 | 100n+10 ---(5) now we can write n+10 | 100(n+10)---(6) ..using (5) and (6) n+10| 100(n+10)-(100n+10) ..that means n+10 | 990 ...and now if we want to get the highest value of n, we have to take the biggest factor of 990 which is 990 itself .so n+10 = 990 (where n has it's biggest value ) so the BIGGEST value of n is 980.🙃

Since the numerator is a cubic, $\frac{n^3+10}{n+10}$ can be written in the form $An^2+Bn+C+\frac{D}{n+10}$ where $A,B,C,D$ are integers. This is because when we do polynomial long division, we are using only addition, subtraction and multiplication which are closed in the integers.

Then $n^3+10 = An^2(n+10) + Bn(n+10) + C(n+10) + D$ which must hold for all $A,B,C,D$ . When $n = -10$ , $-990 = D$ , and so the reciprocal term is $-\frac{990}{n+10}$ . All the other terms are integers because $A, B, C, D, n$ are integers, so $-\frac{990}{n+10}$ must also be an integer.

Thus $n+10 = 990 \Rightarrow n = \boxed{980}$ .