which polynomial has $(3x + 2)$ as a binomial factor?\n$6x^3 + 3x^2 + 4x + 2$\n$12x^2 + 15x + 8x +…

Explanation:

Step1: Find root of $3x+2=0$

$3x+2=0 \implies x=-\frac{2}{3}$

Step2: Test Option 1

Substitute $x=-\frac{2}{3}$ into $6x^3+3x^2+4x+2$: $$6\left(-\frac{2}{3}\right)^3 + 3\left(-\frac{2}{3}\right)^2 + 4\left(-\frac{2}{3}\right) + 2 = 6\left(-\frac{8}{27}\right) + 3\left(\frac{4}{9}\right) - \frac{8}{3} + 2 = -\frac{16}{9} + \frac{4}{3} - \frac{8}{3} + 2 = -\frac{16}{9} - \frac{4}{3} + 2 = -\frac{16}{9} - \frac{12}{9} + \frac{18}{9} = -\frac{10}{9} \neq 0$$

Step3: Test Option 2

First simplify $12x^2+15x+8x+10=12x^2+23x+10$. Substitute $x=-\frac{2}{3}$: $$12\left(-\frac{2}{3}\right)^2 + 23\left(-\frac{2}{3}\right) + 10 = 12\left(\frac{4}{9}\right) - \frac{46}{3} + 10 = \frac{16}{3} - \frac{46}{3} + \frac{30}{3} = 0$$

Step4: Verify (optional)

Factor $12x^2+23x+10$: $$12x^2+23x+10 = 12x^2+15x+8x+10 = 3x(4x+5)+2(4x+5)=(3x+2)(4x+5)$$

Answer:

B. $12x^2 + 15x + 8x + 10$