To solve this problem, we will use the distributive property to multiply the two binomials together. This involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms.
$(7x + 7)(7x^2 - 7x - 7) = 7x(7x^2 - 7x - 7) + 7(7x^2 - 7x - 7)$
$= 49x^3 - 49x^2 - 49x + 49x^2 - 49x - 49$
$= \boxed{49x^3 - 98x - 49}$
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