a. Use synthetic division to show that 2 is a solution of the polynomial equation below.
\[
12 x^{3}-14 x^{2}+17 x-74=0
\]
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.
\[
f(x)=12 x^{3}-14 x^{2}+17 x+23
\]
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, $12 x^{3}-14 x^{2}+17 x-74$ 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 ：\(\boxed{2}\) is a solution to the equation because the remainder of the division, \(12 x^{3}-14 x^{2}+17 x-74\) divided by \(x-2\), is \(\boxed{0}\).

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 \(\boxed{2}\) millimeters.