The area $A(x)$ of a rectangle is given by multiplying its length and width:

$A(x) = L(x) \cdot W(x)$

Given functions:

$L(x) = 4x$ $W(x) = 7x^2 - 4x + 2$

Now, multiply:

$A(x) = (4x) \cdot (7x^2 - 4x + 2)$

Distribute $4x$:

$A(x) = 4x \cdot 7x^2 + 4x \cdot (-4x) + 4x \cdot 2$

$A(x) = 28x^3 - 16x^2 + 8x$

Thus, the correct answer is:

$\boxed{(L \cdot W)(x) = 28x^3 - 16x^2 + 8x}$

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Would you like a breakdown of any specific step?

Here are 5 related questions to expand your understanding:

1. How do you find the perimeter of the rectangle given $L(x)$ and $W(x)$?
2. What happens to the area when $x = 0$ or $x = 1$?
3. How do you determine the domain of the area function $A(x)$?
4. How would you express the area as a function of a different variable, say $t$, if $x = t^2$?
5. How does the rate of change of the area function relate to its derivative?

Tip:

When multiplying polynomials, distribute each term carefully and ensure you combine like terms correctly!