Factor by grouping. \[ 4 w^{3}+3 w^{2}+8 w+6 \]

Step 1 ：The expression given is \(4 w^{3}+3 w^{2}+8 w+6\).

Step 2 ：We can factor this expression by grouping the terms two at a time.

Step 3 ：First, we group the terms as \((4w^{3} + 3w^{2}) + (8w + 6)\).

Step 4 ：Then, we factor out the greatest common factor from each group.

Step 5 ：From the first group, we can factor out \(w^{2}\) to get \(w^{2}(4w + 3)\).

Step 6 ：From the second group, we can factor out 2 to get \(2(4w + 3)\).

Step 7 ：Now, we can see that \((4w + 3)\) is a common factor.

Step 8 ：So, the factored form of the expression is \((4w + 3)(w^{2} + 2)\).

Step 9 ：Final Answer: The factored form of the expression is \(\boxed{(4w + 3)(w^{2} + 2)}\).