If you evaluate your polynomial at x= - 3, the polynomial = 0. So, that means ( x + 3 ) is a factor. Either using long division, or synthetic division, you'll find out that 2x^3 - 9x^2 + 3x + 4 remains. Then if you try x = 4 on that remaining polynomial, it will = 0. Use long division or synthetic division to find out that 2x^2 - x - 1 is left over. That trinomial factors into ( 2x + 1 ) ( x - 1 ).
factored form is ( x+3 )( x - 4 )( 2x + 1 )( x - 1 )