For the function $f(x)=-11 x^{2}+9 x-10$, find the equation of the tangent line at $x=-2$

Step 1 ：First, find the derivative of the function: \(f'(x) = -22x + 9\)

Step 2 ：Next, find the slope of the tangent line at \(x = -2\): \(f'(-2) = -22(-2) + 9 = 44 + 9 = 53\)

Step 3 ：Now, find the point on the function at \(x = -2\): \(f(-2) = -11(-2)^2 + 9(-2) - 10 = -44 - 18 - 10 = -72\)

Step 4 ：The point on the function is \((-2, -72)\) and the slope of the tangent line is 53

Step 5 ：Use the point-slope form of a line: \(y - y_1 = m(x - x_1)\)

Step 6 ：Plug in the point and slope: \(y - (-72) = 53(x - (-2))\)

Step 7 ：Simplify the equation: \(y + 72 = 53(x + 2)\)

Step 8 ：Distribute the 53: \(y + 72 = 53x + 106\)

Step 9 ：Subtract 72 from both sides: \(y = 53x + 34\)

Step 10 ：\(\boxed{y = 53x + 34}\)