Instead of expanding each answer choice, divide $2x^{3}-5x^{2}+4x-9$ by $x-2$ (polynomial long division or synthetic division) to find the quotient and remainder.

Once you find the quotient $Q(x)$ and the remainder $R$, write the original fraction as $Q(x) + \dfrac{R}{x-2}$ and see which choice has the same $Q(x)$ and the same sign on $\dfrac{R}{x-2}$.

In Desmos, type

f(x) = (2x^3 - 5x^2 + 4x - 9) / (x - 2)

This is the function given in the problem.

Type the four options as

• A(x) = 2x^2 - x + 2 + 5/(x - 2)
• B(x) = 2x^2 - x - 5 + 2/(x - 2)
• C(x) = 2x^2 + x + 2 - 5/(x - 2)
• D(x) = 2x^2 - x + 2 - 5/(x - 2)

so they all appear on the same graph.

Zoom out if needed and see which of $A(x)$, $B(x)$, $C(x)$, or $D(x)$ has a graph that lies exactly on top of the graph of $f(x)$ (for all $x$ where they are defined). You can also use a table in Desmos (click the gear icon) and compare numerical values of $f(x)$ and each candidate at several $x$-values (such as $x=0,1,3$); the function that always matches $f(x)$ is the equivalent expression.

We want to rewrite

in a different but mathematically equal form.

When you divide one polynomial by another linear polynomial like $x-2$, you can always write

where $Q(x)$ is the quotient (a polynomial) and $R$ is the remainder (a constant). Our job is to find $Q(x)$ and $R$, then pick the choice that matches $Q(x) + \dfrac{R}{x-2}$.

Because the divisor is $x-2$, use synthetic division with the number $2$.

1. Write the coefficients of the numerator in order:

   • $2x^3$: coefficient $2$
   • $-5x^2$: coefficient $-5$
   • $4x$: coefficient $4$
   • constant $-9$: coefficient $-9$

   So we use: $2,\,-5,\,4,\,-9$.

2. Set up synthetic division with $2$ on the left:

• Bring down the first coefficient: $2$.
• Multiply $2$ (bottom) by $2$ (on the left): $4$. Add to $-5$ to get $-1$.
• Multiply $-1$ by $2$: $-2$. Add to $4$ to get $2$.
• Multiply $2$ by $2$: $4$. Add to $-9$ to get $-5$.

3. The bottom row $2,\,-1,\,2,\,-5$ tells us:
   • The quotient is the polynomial with coefficients $2, -1, 2$, so $Q(x) = 2x^2 - x + 2$.
   • The remainder is $R = -5$.

From polynomial division, we now know

So the expression is

Compare this with the answer choices: it matches choice D, $2x^{2}-x+2-\dfrac{5}{x-2}$.