Which polynomial has (3x+2) as a binomial factor? 6x^3+3x^2+4x+2 12x^2+15x+8x+10 18x^3-12x^2+9x-6 21x^4+7x^3+6x+2

To determine which polynomial has $(3x+2)$ as a binomial factor, we can use polynomial division or synthetic division. Let's perform synthetic division for each polynomial to see which one has $(3x+2)$ as a factor.<br /><br />### Polynomial 1: $6x^{3}+3x^{2}+4x+2$<br /><br />Let's divide by $(3x+2)$ using synthetic division:<br /><br />1. Rewrite $3x+2$ as $3x - (-2)$.<br />2. Use $-2$ for synthetic division.<br /><br />```<br />-2 | 6 3 4 2<br /> | -12 18 -28<br /> -------------------<br /> 6 -9 22 -26<br />```<br /><br />The remainder is not zero, so $(3x+2)$ is not a factor of $6x^{3}+3x^{2}+4x+2$.<br /><br />### Polynomial 2: $12x^{2}+15x+8x+10$<br /><br />This polynomial is not correctly written. Let's assume it is $12x^{2}+23x+10$.<br /><br />Let's divide by $(3x+2)$ using synthetic division:<br /><br />```<br />-2 | 12 23 10<br /> | -24 -14<br /> -------------------<br /> 12 -1 -4<br />```<br /><br />The remainder is not zero, so $(3x+2)$ is not a factor of $12x^{2}+23x+10$.<br /><br />### Polynomial 3: $18x^{3}-12x^{2}+9x-6$<br /><br />Let's divide by $(3x+2)$ using synthetic division:<br /><br />```<br />-2 | 18 -12 9 -6<br /> | -36 96 -174<br /> -------------------<br /> 18 -48 105 -180<br />```<br /><br />The remainder is not zero, so $(3x+2)$ is not a factor of $18x^{3}-12x^{2}+9x-6$.<br /><br />### Polynomial 4: $21x^{4}+7x^{3}+6x+2$<br /><br />Let's divide by $(3x+2)$ using synthetic division:<br /><br />```<br />-2 | 21 7 0 6 2<br /> | -42 -14 28 -108<br /> -----------------------<br /> 21 -35 14 34 -106<br />```<br /><br />The remainder is not zero, so $(3x+2)$ is not a factor of $21x^{4}+7x^{3}+6x+2$.<br /><br />None of the given polynomials have $(3x+2)$ as a binomial factor.