Answer :

We know that if (x – a) is a zero of a polynomial then (x – a) is a factor of f(x)

Since –2 and –1 are zeros of f(x).Therefore, (x + 2)(x + 1) = x^{2} + 3x + 2 is a factor of f(x).

Now on dividing f(x) = 2x^{4} + x^{3} – 14x^{2} –19x – 6 by g(x) = x^{2} + 3x + 2 to find other zeros.

By applying division algorithm, we have:

2x^{4} + x^{3} – 14x^{2} –19x – 6 = (x^{2} + 3x + 2)(2x^{2} – 5x – 3)

2x^{4} + x^{3} – 14x^{2} –19x – 6 = (x + 2)(x + 1)(2x^{2} – 5x – 3)

2x^{4} + x^{3} – 14x^{2} –19x – 6 = (x + 2)(x + 1)(2x + 1)(x – 3)

**Hence, the zeros of the given polynomial are –1/2, –1, –2, 3.**

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