Let p(x) = 30 x^3 -7 x^2 - 7 x + 2. Prove that (2x + 1) is a factor of p(x) and factorise p(x) completely.

Using the Factor Theorem, we know that (2x + 1) = 2(x + 1/2) = 2(x - (-1/2) ) is a factor of p(x) if and only if p(-1/2) = 0, which is true. Now that we know a factor of p(x), we use the polynomial division method to find p(x) = (2x + 1)(15 x^2 - 11x + 2) and factorise the quadratic with some simple algebra 15 x^2 - 11x + 2 = 15 x^2 - 5x - 6x + 2 = 5x(3x - 1) - 2(3x - 1) = (3x - 1)(5x - 2), which gives us the final answer: p(x) = (2x + 1)(3x - 1)(5x - 2).