We are given a polynomial $x^3 - 2x^2 + mx + 4$ and told that when it is divided by $x-3$, the remainder is $-2$. We need to find the value of $m$.

1. Problem Description

We are given a polynomial $x^3 - 2x^2 + mx + 4$ and told that when it is divided by $x-3$, the remainder is $-2$. We need to find the value of $m$.

2. Solution Steps

We can use the Remainder Theorem, which states that if a polynomial $P(x)$ is divided by $x-c$, the remainder is $P(c)$. In this case, $P(x) = x^3 - 2x^2 + mx + 4$ and we are dividing by $x-3$, so $c=3$. The remainder is given as $-2$. Therefore, we have $P(3) = -2$.

Substitute $x=3$ into the polynomial:

$P(3) = (3)^3 - 2(3)^2 + m(3) + 4$

$P(3) = 27 - 2(9) + 3m + 4$

$P(3) = 27 - 18 + 3m + 4$

$P(3) = 9 + 3m + 4$

$P(3) = 13 + 3m$

Now we know that $P(3) = -2$, so we can set up the equation:

$13 + 3m = -2$

Subtract 13 from both sides:

$3m = -2 - 13$

$3m = -15$

Divide by 3:

$m = \frac{-15}{3}$

$m = -5$

3. Final Answer

$m = -5$