Find all zeros of the function and write the polynomial as a product of linear factors. f(x) = 3x4 + 4x3 + 13x2 + 16x + 4

Don't try anything positive.
f(-1) = 3 - 4 + 13 - 16 + 4 = 0 This one works so one factor is (x + 1)
f(-1/3) = gives a 0. So another factor is 3x + 1
The last factor is found by dividing 3x^2 + 4x + 1 into the original equation. That's (3x + 1)(x + 1) expanded.
When you do that you find that the last factor is (x^2 + 1)
You can see how this is factored at Wolframalpha dot com