Find [tex]$(f \cdot g)(x)$[/tex] if [tex]$f(x) = 7x^3 - 5x^2 + 42x - 30$[/tex] and [tex]$g(x) = 7x - 5$[/tex].

A. [tex]$(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150$[/tex]

B. [tex]$(f \cdot g)(x) = 49x^4 - 269x^2 - 150$[/tex]

C. [tex]$(f \cdot g)(x) = 49x^4 + 269x^2 + 150$[/tex]

D. [tex]$(f \cdot g)(x) = 49x^4 + 70x^3 - 319x^2 + 420x - 150$[/tex]

We start with
[tex]$$
f(x) = 7x^3 - 5x^2 + 42x - 30 \quad \text{and} \quad g(x) = 7x - 5.
$$[/tex]

The product is given by
[tex]$$
(f \cdot g)(x) = f(x) \cdot g(x) = \Bigl(7x^3 - 5x^2 + 42x - 30\Bigr)(7x - 5).
$$[/tex]

We will multiply each term of [tex]$f(x)$[/tex] by the entire expression [tex]$7x - 5$[/tex].

1. Multiply the first term [tex]$7x^3$[/tex] by [tex]$7x - 5$[/tex]:
[tex]$$
7x^3 \cdot (7x - 5) = 7x^3 \cdot 7x - 7x^3 \cdot 5 = 49x^4 - 35x^3.
$$[/tex]

2. Multiply the second term [tex]$-5x^2$[/tex] by [tex]$7x - 5$[/tex]:
[tex]$$
-5x^2 \cdot (7x - 5) = -5x^2 \cdot 7x + (-5x^2) \cdot (-5) = -35x^3 + 25x^2.
$$[/tex]

3. Multiply the third term [tex]$42x$[/tex] by [tex]$7x - 5$[/tex]:
[tex]$$
42x \cdot (7x - 5) = 42x \cdot 7x - 42x \cdot 5 = 294x^2 - 210x.
$$[/tex]

4. Multiply the fourth term [tex]$-30$[/tex] by [tex]$7x - 5$[/tex]:
[tex]$$
-30 \cdot (7x - 5) = -30 \cdot 7x + (-30) \cdot (-5) = -210x + 150.
$$[/tex]

Now, combine all the results:
[tex]$$
\begin{array}{rcl}
(f \cdot g)(x) &=& \phantom{+}49x^4 - 35x^3 \\
&& +\; (-35x^3 + 25x^2) \\
&& +\; (294x^2 - 210x) \\
&& +\; (-210x + 150).
\end{array}
$$[/tex]

Group like terms:

- For [tex]$x^4$[/tex]:
[tex]$$
49x^4.
$$[/tex]
- For [tex]$x^3$[/tex]:
[tex]$$
-35x^3 - 35x^3 = -70x^3.
$$[/tex]
- For [tex]$x^2$[/tex]:
[tex]$$
25x^2 + 294x^2 = 319x^2.
$$[/tex]
- For [tex]$x$[/tex]:
[tex]$$
-210x - 210x = -420x.
$$[/tex]
- Constant term:
[tex]$$
150.
$$[/tex]

Thus, the product simplifies to
[tex]$$
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150.
$$[/tex]

This matches the answer:
[tex]$$
\boxed{49x^4 - 70x^3 + 319x^2 - 420x + 150.}
$$[/tex]