Which of the following is a factor of the polynomial \(6x^{4} + 2x^{3} + 15x + 5\)?

To determine if a given expression is a factor of a polynomial, we can use the factor theorem which states that if a polynomial f(a) is divided by x - a, then the remainder is zero if and only if x - a is a factor of the polynomial. To use the factor theorem in this case, we can try to divide the polynomial \(6x^{4} + 2x^{3} + 15x + 5\) by each of the given expressions to see if the remainder is zero. We can use long division or synthetic division to perform the division. Using synthetic division, we can test each option as follows: - For 3x + 1: -1/3 | 6 2 0 15 5 | -2 2 -5 -------------- 6 0 2 10 0 The remainder is zero, so 3x + 1 is a factor of the polynomial. - For x + 1: -1 | 6 2 0 15 5 | -6 -8 -7 ------------ 6 -4 -8 8 The remainder is not zero, so x + 1 is not a factor of the polynomial. - For 2x + 1: -1/2 | 6 2 0 15 5 | -3 -2.5 ------------ 6 0 3 12.5 The remainder is not zero, so 2x + 1 is not a factor of the polynomial. - For x + 2: -2 | 6 2 0 15 5 | -12 -20 -30 ------------- 6 -10 -20 -15 The remainder is not zero, so x + 2 is not a factor of the polynomial. Therefore, only 3x + 1, is a factor of the polynomial \(6x^{4} + 2x^{3} + 15x + 5\). The answer is: 3x + 1.