### Instruction: Find the smallest integer value of $n$ such that $n^2-5n-14$ is negative.### Response: I notice that $n^2-5n-14$ is a quadratic expression, and I recall that a quadratic function has a negative value between its two roots.
To find the roots of the quadratic expression, I can try to factor it.
I look for two numbers that multiply to -14 and add to -5.
I see that -7 and 2 are such numbers, so I can write $n^2-5n-14 = (n-7)(n+2)$.
Now I can see that the roots of the quadratic expression are $n=7$ and $n=-2$.
This means that the expression is negative when $-2 < n < 7$.
The smallest integer value of $n$ that satisfies this inequality is $n=-1$.
I can check that this is correct by plugging in $n=-1$ into the expression and seeing that it is indeed negative: $(-1)^2-5(-1)-14 = 1+5-14 = -8$.