We are given a polynomial $f(x) = x^3 + mx^2 + x + 6$. We are told that when $f(x)$ is divided by $(x-6)$, the remainder is 84. We need to find the value of $m$.

1. Problem Description

We are given a polynomial $f(x) = x^3 + mx^2 + x + 6$. We are told that when $f(x)$ is divided by $(x-6)$, the remainder is

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4. We need to find the value of $m$.

2. Solution Steps

By the Remainder Theorem, when a polynomial $f(x)$ is divided by $(x-c)$, the remainder is $f(c)$. In our case, $c = 6$, so the remainder is $f(6)$. We are given that the remainder is 84, so $f(6) = 84$.

Now, let's evaluate $f(6)$:

$f(6) = (6)^3 + m(6)^2 + (6) + 6$

$f(6) = 216 + 36m + 6 + 6$

$f(6) = 228 + 36m$

Since $f(6) = 84$, we can set up the equation:

$228 + 36m = 84$

Now we solve for $m$:

$36m = 84 - 228$

$36m = -144$

$m = \frac{-144}{36}$

$m = -4$

3. Final Answer

The value of $m$ is -4.