We are given the following function:

Part A. We are asked to determine the slope of the tangent line at "x = 4".
To determine the slope of the tangent line at a point "x = a" we use the following limit definition:

Now, we substitute the value of "x = 4" in the limit definition, we get:

Now, we determine the value of f(4) from the given function;

Solving the operations:

Now, we substitute the values in the limit:

Now, we take 7 as a common factor:

Now, we factor de denominator, we get:

Now, we cancel out the "x - 4":

Now, we substitute the values of "x = 4", we get:

Therefore, the slope is 49.
Part B. We are asked to determine the equation of the tangent line. We have that the general form of a line equation is:

From part A we have that the slope is 49, therefore, we have:

Now, we need to determine a point on the line to determine the value of "b". We already have that the point "x = 4" is part of the line. To determine the corresponding value of "y" we substitute in the given function:

Solving we get:

Therefore, the point (x, y) = (4, 84) is on the line. Substituting we get:

Solving the product:

Now, we subtract 196 from both sides:

Now, we substitute the value in the line equation:

Thus we have determined the equation of the tangent line.