First, use the fact that the polynomial is divisible by $x+2$ to find an equation involving $a$ and $b$. Then, use the given remainder when dividing by $x-3$ to form a second equation.

After you have the two equations in $a$ and $b$, choose elimination or substitution to solve the system. Once you find $a$ and $b$, be careful to compute $a-b$, not $a+b$ or just $a$ or $b$.

In Desmos, use $x$ for $a$ and $y$ for $b$. Enter the two equations as

• 2x - y = -6
• 3x + y = -17 These are two lines whose intersection gives $(x,y) = (a,b)$.

Tap the point where the two lines intersect; Desmos will show its coordinates $(x,y)$. These are the values of $a$ and $b$.

In a new expression line, type x - y and then substitute the intersection’s $x$ and $y$ values (or define parameters for $a$ and $b$ using those values). The resulting value shown by Desmos is the required $a-b$.

If a polynomial $f(x)$ is divisible by $x+2$, then $x=-2$ is a root and $f(-2)=0$ (this is the Remainder Theorem).

Here $f(x)=x^3+ax^2+bx+20$.

Compute $f(-2)$:

Set this equal to $0$:

Divide by $2$ to simplify:

so

When a polynomial $f(x)$ is divided by $x-3$, the remainder is $f(3)$.

We are told this remainder is $-4$, so $f(3) = -4$.

Compute $f(3)$:

Set this equal to $-4$:

Subtract $47$ from both sides:

Divide by $3$ to simplify:

Now solve the system:

Add the two equations to eliminate $b$:

so

Substitute $a$ into $3a + b = -17$:

Add $\tfrac{69}{5}$ to both sides. Since $-17 = -\tfrac{85}{5}$,

so

We now have

Compute $a - b$:

So the value of $a - b$ is $-\tfrac75$. This corresponds to choice D.