Part 1 of 2 Use synthetic division and the factor theorem to determine whether $x-2$ is a factor of $f(x)$. \[ f(x)=4 x^{3}-10 x^{2}+7 x-6 \] Complete the first row of the synthetic division table.

Step 1 ：Define the polynomial \(f(x)=4 x^{3}-10 x^{2}+7 x-6\).

Step 2 ：Use synthetic division to determine whether \(x-2\) is a factor of \(f(x)\).

Step 3 ：Start by writing the coefficients of the polynomial in the first row of the synthetic division table: [4, -10, 7, -6].

Step 4 ：Perform the synthetic division process: Multiply the root (2) by the first coefficient (4), add the result to the second coefficient (-10), and write the result (-2) in the second row of the table. Repeat this process for the remaining coefficients.

Step 5 ：The second row of the synthetic division table is [4, -2, 3, 0].

Step 6 ：The remainder is the last element of the second row, which is 0.

Step 7 ：Since the remainder is 0, \(x-2\) is a factor of \(f(x)\).

Step 8 ：\(\boxed{x-2 \text{ is a factor of } f(x)}\)