Let $$f: R \rightarrow R$$ be defined $$f(x) = ae^{2x} + be^x + cx$$. If $$f(0) = -1$$, $$f'(\log_e 2) = 21$$ and $$\int_0^{\log 4}(f(x) - cx) \, dx = \frac{39}{2}$$, then the value of $$|a + b + c|$$ equals:

$$f(x) = ae^{2x} + be^x + cx$$. $$f(0) = a + b = -1$$ ... (1)

$$f'(x) = 2ae^{2x} + be^x + c$$. 

$$f'(\ln 2) = 2a(4) + b(2) + c = 8a + 2b + c = 21$$ ... (2)

$$\int_0^{\ln 4}(f(x)-cx)dx = \int_0^{\ln 4}(ae^{2x}+be^x)dx = \left[\frac{a}{2}e^{2x}+be^x\right]_0^{\ln 4}$$

$$= \frac{a}{2}(16) + b(4) - \frac{a}{2} - b = \frac{15a}{2} + 3b = \frac{39}{2}$$ ... (3)

From (3): $$15a + 6b = 39 \Rightarrow 5a + 2b = 13$$ ... (3')

From (1): $$b = -1 - a$$. Substituting in (3'): $$5a + 2(-1-a) = 13 \Rightarrow 3a = 15 \Rightarrow a = 5$$.

$$b = -6$$. From (2): $$40 - 12 + c = 21 \Rightarrow c = -7$$.

$$|a + b + c| = |5 - 6 - 7| = |-8| = 8$$.