If the $$x$$-intercept of a focal chord of the parabola $$y^2 = 8x + 4y + 4$$ is 3, then the length of this chord is equal to _____.

$$y^2 - 4y = 8x + 4 \implies (y-2)^2 = 8(x+1)$$

Comparing with $$(y-k)^2 = 4a(x-h)$$: $$h = -1, \quad k = 2, \quad 4a = 8 \implies a = 2$$

The focus of the parabola is $$(h+a, k)$$: $$S = (-1 + 2, 2) = (1, 2)$$

Since the chord passes through the focus $$S(1,2)$$ and has $$x$$-intercept $$3$$ (passes through $$A(3,0)$$):

$$m = \frac{0 - 2}{3 - 1} = -1 \implies \tan\alpha = -1 \implies \alpha = 135^\circ$$

$$\text{Length} = 4a\csc^2\alpha = 8\csc^2(135^\circ) = 8(\sqrt{2})^2 = 16$$