Evaluate the given definite integral. \[ \int_{-3}^{-2} 2 x\left(7-x^{2}\right)^{3} d x \] $\int_{-3}^{-2} 2 x\left(7-x^{2}\right)^{3} d x=\square($ Type an integer or a simplified fraction.)

Step 1 ：First, let's perform the u-substitution. We'll set \(u = 7 - x^2\), so \(du = -2x dx\). This means our integral becomes \(-\int u^3 du\) from \(u = 7 - (-3)^2 = 2\) to \(u = 7 - (-2)^2 = 3\).

Step 2 ：Now we can find the antiderivative of \(u^3\), which is \((1/4)u^4\).

Step 3 ：We'll evaluate this at 3 and 2 and subtract the two values to find the definite integral.

Step 4 ：The definite integral of the function from -3 to -2 is \(65/4\).

Step 5 ：Final Answer: \(\boxed{\frac{65}{4}}\)