Factor the trinomial by grouping. \[ 3 z^{2}+2 z+4 \]

Step 1 ：Given the trinomial \(3 z^{2}+2 z+4\).

Step 2 ：This is a quadratic trinomial. Factoring by grouping is a method used when the expression has four terms. However, in this case, the expression has only three terms. Therefore, we can't factor this trinomial by grouping.

Step 3 ：Instead, we can try to factor it using the quadratic formula or by completing the square. However, it's also possible that the trinomial can't be factored using integers or rational numbers.

Step 4 ：The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where a, b, and c are the coefficients of the quadratic trinomial. In this case, a = 3, b = 2, and c = 4.

Step 5 ：Substitute these values into the quadratic formula and see if we get real numbers. If we do, then the trinomial can be factored. If not, then it can't be factored using integers or rational numbers.

Step 6 ：The solutions are complex numbers, which means that the trinomial can't be factored using integers or rational numbers.

Step 7 ：\(\boxed{\text{The trinomial } 3 z^{2}+2 z+4 \text{ is prime and can't be factored.}}\)