unction $y=\left(\frac{3 x^{2}-5}{2 x+1}\right)^{6}$ find $\frac{d y}{d x}$. Factor your

Step 1 ：The problem is asking for the derivative of the function \(y=\left(\frac{3 x^{2}-5}{2 x+1}\right)^{6}\).

Step 2 ：To find the derivative, we can use the chain rule and the quotient rule. The chain rule is a formula to compute the derivative of a composition of functions. In this case, our outer function is \(u^6\) and our inner function is \(\frac{3 x^{2}-5}{2 x+1}\). The quotient rule is a formula for finding the derivative of a quotient of functions. In this case, our numerator function is \(3x^2 - 5\) and our denominator function is \(2x + 1\).

Step 3 ：The derivative of the function \(y=\left(\frac{3 x^{2}-5}{2 x+1}\right)^{6}\) is \(36x(3x^2 - 5)^5/(2x + 1)^6 - 12(3x^2 - 5)^6/(2x + 1)^7\).

Step 4 ：However, the question asks to factor the derivative. We can see that both terms in the derivative have common factors of \((3x^2 - 5)^5/(2x + 1)^6\). We can factor this out to simplify the derivative.

Step 5 ：The derivative of the function \(y=\left(\frac{3 x^{2}-5}{2 x+1}\right)^{6}\), factored, is \(\frac{(3x^2 - 5)^5(-36x^2 + 36x(2x + 1) + 60)}{(2x + 1)^7}\).

Step 6 ：\(\boxed{\frac{(3x^2 - 5)^5(-36x^2 + 36x(2x + 1) + 60)}{(2x + 1)^7}}\) is the final answer.