a. Use synthetic division to show that 2 is a solution of the polynomial equation below.
$$12x^3-14x^2+17x-74=0$$
b. Use the solution from part (a) to solve this problem. The number of eggs, $f(x)$, in a female moth is a function of her abdominal width, in millimeters, modeled by the equation below.
$$f(x)=12x^3-14x^2+17x+23$$
What is the abdominal width when there are 97 eggs?
a. The number 2 is a solution to the equation because the remainder of the division, $12x^3-14x^2+17x-74$ divided by $x-2$, is
b. The abdominal width is millimeters.

Step 1 ：Perform synthetic division on the polynomial $12x^3-14x^2+17x-74$ by $x-2$. If the remainder is zero, then 2 is a solution of the polynomial equation.

Step 2 ：The remainder of the synthetic division is zero, which confirms that 2 is a solution of the polynomial equation.

Step 3 ：$\boxed{{\displaystyle 2}}$ is a solution to the equation because the remainder of the division, $12x^3-14x^2+17x-74$ divided by $x-2$, is $\boxed{{\displaystyle 0}}$.

Step 4 ：Solve the equation $f(x)=97$ for $x$. This means we need to solve the equation $12x^3-14x^2+17x+23=97$ for $x$.

Step 5 ：The solutions to the equation $f(x)=97$ are $2$, $-5/12-\sqrt{419}i/12$, and $-5/12+\sqrt{419}i/12$. Since the abdominal width cannot be negative or complex, the only valid solution is $2$.

Step 6 ：The abdominal width is $\boxed{{\displaystyle 2}}$ millimeters.