The linear approximation, L(x), of f(x) at (1 , -8) is just the equation of the tangent line to f(x) at that point. It is called that because for x-values close to x = 1, the y-value of the tangent line will be a decent approximation of the y-value of f(x).
To find the equation for L(x), we just need the slope of the tangent line at x = 1, which we find by taking the derivative of f(x), and evaluating it at x = 1:
f'(x) = 3x2 + 10x - 9
f'(1) = 4
y + 8 = 4 (x - 1)
L(x) = 4x - 12