Context: The functions $f(x) = x + 3$ and $g(x) = 7x^2 - rx + 63$ are given, where $r$ is a constant, and their product satisfies $f(x) \cdot g(x) = 7x^3 + 189$ for all $x$. We need $r$.

Answer: 21

Why: Expand the product: $(x + 3)(7x^2 - rx + 63) = 7x^3 - rx^2 + 63x + 21x^2 - 3rx + 189$, which collects to $7x^3 + (21 - r)x^2 + (63 - 3r)x + 189$. For this to equal $7x^3 + 189$, the $x^2$ and $x$ coefficients must both be $0$: $21 - r = 0$ and $63 - 3r = 0$. Each equation gives $r = 21$.

Verification: With $r = 21$: $(x + 3)(7x^2 - 21x + 63) = 7(x + 3)(x^2 - 3x + 9) = 7(x^3 + 27) = 7x^3 + 189$, the sum-of-cubes pattern.

Common slips: Distributing the $3$ into only some terms of $g(x)$, or a sign error when solving $21 - r = 0$ that produces $-21$.