Ben is trying to determine if x = 2/3 is a solution to the quadratic equation 9x^2 + 3x - 5 = 6. Which explanation below demonstrates the correct reasoning?

To verify if \( x = \frac{2}{3} \) is a solution to the equation \( 9x^2 + 3x - 5 = 6 \), we can substitute \( x = \frac{2}{3} \) into the left-hand side of the equation and check if it equals 6.

First, calculate \( 9x^2 + 3x - 5 \) with \( x = \frac{2}{3} \):

1. Calculate \( 9x^2 \):

\[
9 \left( \frac{2}{3} \right)^2 = 9 \cdot \frac{4}{9} = 4
\]

2. Calculate \( 3x \):

\[
3 \left( \frac{2}{3} \right) = 2
\]

3. Now sum these results and subtract 5:

\[
9x^2 + 3x - 5 = 4 + 2 - 5 = 1
\]

Now we compare the left-hand side to the right-hand side of the original equation:

\[
1 \neq 6
\]

Since \( 1 \) does not equal \( 6 \), we conclude that \( x = \frac{2}{3} \) is not a solution to the equation.

Looking at the options:

- A states: "Yes, x = 2/3 is a solution because substituting it back into the equation results in 1=6 which is a true statement" (incorrect).
- B states: "Yes, x = 2/3 is a solution because substituting it back into the equation results in 6=6 which is a true statement" (incorrect).
- C states: "No, x = 2/3 is not a solution because substituting it back into the equation results in 1=6 which is a false statement" (correctly identifies that 1 does not equal 6).
- D states: "No, x = 2/3 is not a solution because substituting it back into the equation results in 6=6 which is a false statement" (incorrect).

The correct reasoning is given by option **C**.