Factor. \[ 3 z^{2}+z-14 \]

Step 1 ：We are given the quadratic equation \(3 z^{2}+z-14\).

Step 2 ：We need to find two numbers that multiply to -42 (which is the product of 3 and -14) and add to 1 (the coefficient of z).

Step 3 ：The two numbers that satisfy these conditions are 7 and -6.

Step 4 ：We can rewrite the middle term of the quadratic equation as the sum of the terms 7z and -6z. So, the equation becomes \(3 z^{2}+7z-6z-14\).

Step 5 ：We can now factor by grouping. The first two terms can be grouped together, and the last two terms can be grouped together.

Step 6 ：The factored form of the equation is \((z - 2)(3z + 7)\).

Step 7 ：\(\boxed{(z - 2)(3z + 7)}\) is the final answer.