Question 763522
Your possible values to check according to Rational Roots Theorem with synthetic division are PLUS AND MINUS OF these: { 5/3, 1/3,  5, 3, 1 }.


Greatly abbreviating the process here, You may find something about like this:

Checking 5, remainder 0, quotient {{{3x^2+8x-3}}}.
Checking -3, remainder 0, quotient {{{3x-1}}}.


The polynomial in factored form is {{{highlight((x-5)(x+3)(3x-1))}}}.
The next zero based on the last binomial factor is +{{{1/3}}}, so the three roots or zeros are  {{{-3}}}, {{{5}}}, {{{1/3}}}.


These are all Real zeros.  They are also Complex zeros for which the imaginary compenents are zero.