Choice C is correct. Multiplying the given functions $f$ and $g$ yields $f (x) \cdot g (x) = (x^{2} + b x) (9 x^{2} - 27 x)$ . Applying the distributive property to the right-hand side of this equation yields $f (x) \cdot g (x) = (x^{2}) (9 x^{2} - 27 x) + (b x) (9 x^{2} - 27 x)$ . Applying the distributive property once again to the right-hand side of this equation yields $f (x) \cdot g (x) = (x^{2}) (9 x^{2}) - (x^{2}) (27 x) + (b x) (9 x^{2}) - (b x) (27 x)$ , which is equivalent to $f (x) \cdot g (x) = 9 x^{4} - 27 x^{3} + 9 b x^{3} - 27 b x^{2}$ . Factoring out $x^{3}$ from the second and third terms yields $f (x) \cdot g (x) = 9 x^{4} + (- 27 + 9 b) x^{3} - 27 b x^{2}$ . Since the left-hand sides of $f (x) \cdot g (x) = 9 x^{4} + (- 27 + 9 b) x^{3} - 27 b x^{2}$ and $f (x) \cdot g (x) = 9 x^{4} - 26 x^{3} - 3 x^{2}$ are equal, it follows that $(- 27 + 9 b) x^{3} = - 26 x^{3}$ , or $- 27 + 9 b = - 26$ , and $- 27 b x^{2} = - 3 x^{2}$ , or $- 27 b = - 3$ . Adding $27$ to each side of $- 27 + 9 b = - 26$ yields $9 b equals 1$ . Dividing each side of this equation by $9$ yields $b = \frac{1}{9}$ . Similarly, dividing each side of $- 27 b = - 3$ by $negative 27$ yields $b = \frac{1}{9}$ . Therefore, the value of $b$ is $\frac{1}{9}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.