To find the value of \(m\) and the coordinates of \(P\), we equate the line and the curve:

\(mx + 14 = \frac{12}{x} + 2\)

Multiply through by \(x\) to eliminate the fraction:

\(mx^2 + 14x = 12 + 2x\)

Rearrange to form a quadratic equation:

\(mx^2 + 12x - 12 = 0\)

Since the line is tangent to the curve, the discriminant of this quadratic must be zero:

\(b^2 - 4ac = 0\)

Substitute \(a = m\), \(b = 12\), \(c = -12\):

\(12^2 - 4m(-12) = 0\)

\(144 + 48m = 0\)

\(48m = -144\)

\(m = -3\)

Substitute \(m = -3\) back into the equation to find \(x\):

\(-3x^2 + 12x - 12 = 0\)

Factor or solve for \(x\):

\(x = 2\)

Substitute \(x = 2\) into the original curve equation to find \(y\):

\(y = \frac{12}{2} + 2 = 6 + 2 = 8\)

Thus, the coordinates of \(P\) are \((2, 8)\).