After you clear the denominator, you will get $(x^{2}-x+1)(x-2)+5$ on one side. Expand this product step by step and then simplify.

Once both sides are written as standard polynomials in $x$, compare the coefficients of $x$ and the constant terms on both sides to solve for $k$ and $m$.

After you find $k$ and $m$, remember the problem asks for $k+m$, not $k$ or $m$ separately.

In one line, enter f(x) = (x^3 - 3x^2 + kx + m)/(x - 2) and let Desmos create sliders for k and m. In another line, enter g(x) = x^2 - x + 1 + 5/(x - 2).

Adjust the sliders for k and m until the graphs of f(x) and g(x) overlap everywhere they are both defined (except at $x=2$ where there is a hole). The slider values at that point are the correct $k$ and $m$.

After you have the correct k and m from the sliders, verify that the graphs still coincide for several different $x$-values, then add those two values (by hand or using Desmos) to find $k+m$.

The equation

is stated to hold for all real $x\ne 2$, so we can safely multiply both sides by $x-2$:

Now both sides are polynomials in $x$.

Expand $(x^{2}-x+1)(x-2)$:

Now add the $5$:

So the equation becomes

Because these polynomials are equal for all real $x$, their coefficients for each power of $x$ must match.

Compare the coefficients:

• Coefficient of $x^{3}$: $1=1$ (already matches).
• Coefficient of $x^{2}$: $-3=-3$ (already matches).
• Coefficient of $x$: $k=3$.
• Constant term: $m=3$.

So $k=3$ and $m=3$.

The question asks for $k+m$.

So the correct answer is $6$.