The root of a polynomial P(x) is a number that makes P(x) equal to 0.

Now we are given P(x) = 3x^3 - 10x^2 - 5x.

If we substitute -2/3 for x in P(x), we get

P (-2/3) = 3*(-2/3) ^3 – 10 *(-2/3) ^2 – 5*(-2/3)

= 3*(-8 / 27) – 10*(4/9) – 5*(-2/3)

= -8 / 9 – 40/9 + 20/3

= -18 / 9

= -2

We see that -2 is not equal to zero.

**Therefore -2/3 is not a root of 3x^3 - 10x^2 - 5x.**

P(x) = 3x^3-10x^2-5x.

To examine whether P(x) has a root -2/3.

If the root of P(x) is -2/3, then P(-2/3) = 0 by remainder theorem.

=> P(-2/3) = 3(-2/3)^3-10(-2/3)^2 -5(-2/3) should be zero.

=>3(-8/27) +10(4/9) - 10/3 should be zero.

=> -8/9 +40/9 +10/3 should be zero.

=> (1/9){-8 +40 +30) = 62/9 should be zero. This is a contadiction.

We arrived at a contradiction . So (-2/3) is not a root of P(x).