Let $$a$$ be an integer such that all the real roots of the polynomial $$2x^5 + 5x^4 + 10x^3 + 10x^2 + 10x + 10$$ lie in the interval $$(a, a+1)$$. Then, $$|a|$$ is equal to ______.

Let $$f(x) = 2x^5 + 5x^4 + 10x^3 + 10x^2 + 10x + 10$$. We compute $$f'(x) = 10x^4 + 20x^3 + 30x^2 + 20x + 10 = 10(x^4 + 2x^3 + 3x^2 + 2x + 1) = 10(x^2 + x + 1)^2$$.

Since the discriminant of $$x^2 + x + 1$$ is $$1 - 4 = -3 < 0$$, we have $$x^2 + x + 1 > 0$$ for all real $$x$$. Therefore $$f'(x) = 10(x^2 + x + 1)^2 > 0$$ for all real $$x$$, meaning $$f$$ is strictly increasing. This implies $$f$$ has exactly one real root.

Evaluating: $$f(-2) = 2(-32) + 5(16) + 10(-8) + 10(4) + 10(-2) + 10 = -64 + 80 - 80 + 40 - 20 + 10 = -34 < 0$$.

$$f(-1) = -2 + 5 - 10 + 10 - 10 + 10 = 3 > 0$$.

Since $$f(-2) < 0$$ and $$f(-1) > 0$$, the unique real root lies in $$(-2, -1)$$. Thus $$a = -2$$ and $$|a| = 2$$.