Once both sides have the same denominator, focus on the numerators: what equation do you get if you set $a(x+4)+b(x-5)$ equal to $3x+7$?

After you expand $a(x+4)+b(x-5)$, match the coefficient of $x$ and the constant term with those in $3x+7$ to form two equations in $a$ and $b$. Solve that system, then multiply the values you get for $a$ and $b$.

Use the equations from matching coefficients, $a + b = 3$ and $4a - 5b = 7$. In Desmos, treat $a$ as $x$ and $b$ as $y$, and enter the lines y = 3 - x and y = (4x - 7)/5.

Click on the point where the two lines intersect; Desmos will display its coordinates $(x, y)$. Here, the $x$-coordinate is the value of $a$ and the $y$-coordinate is the value of $b$.

In a new expression line, type the $x$- and $y$-coordinates from the intersection (your $a$ and $b$ values) multiplied together, for example 2.44*0.56 but using the actual numbers you see. Use the result as $ab$, and if needed, use Desmos’s fraction conversion tool to match it to one of the answer choices.

Rewrite the left-hand side over the common denominator $(x-5)(x+4)$.

Now the equation becomes

Because the denominators are the same and $x\ne 5,-4$, the numerators must be equal for all $x$:

Expand the left side:

So we have

Match coefficients of like terms:

• Coefficient of $x$: $a+b = 3$.
• Constant term: $4a - 5b = 7$.

This gives a system of two equations in $a$ and $b$:

From $a+b=3$, express $a$ in terms of $b$:

Substitute into $4a - 5b = 7$:

Simplify:

Solve for $b$:

Now find $a$ using $a = 3 - b$:

Now multiply $a$ and $b$:

So the value of $ab$ is $\frac{110}{81}$, which corresponds to choice A.