We are tasked with finding the difference between two functions $f(x) = -4x(12x^2 - 5x - 11)$ and $g(x) = 54x^3 - 36x^2 + 43x + 55$.

Step 1: Simplify $f(x)$

We begin by expanding $f(x)$:

$f(x) = -4x(12x^2 - 5x - 11)$

Distribute $-4x$ across the terms inside the parentheses:

$f(x) = -4x \cdot 12x^2 - 4x \cdot (-5x) - 4x \cdot (-11)$

$f(x) = -48x^3 + 20x^2 + 44x$

So, the simplified form of $f(x)$ is:

$f(x) = -48x^3 + 20x^2 + 44x$

Step 2: Write the difference $f(x) - g(x)$

Now, we subtract $g(x)$ from $f(x)$:

$f(x) - g(x) = (-48x^3 + 20x^2 + 44x) - (54x^3 - 36x^2 + 43x + 55)$

Step 3: Simplify the difference

Distribute the negative sign through the terms of $g(x)$:

$f(x) - g(x) = -48x^3 + 20x^2 + 44x - 54x^3 + 36x^2 - 43x - 55$

Now, combine like terms:

• For $x^3$: $-48x^3 - 54x^3 = -102x^3$
• For $x^2$: $20x^2 + 36x^2 = 56x^2$
• For $x$: $44x - 43x = 1x$
• For the constant term: $-55$

Thus, the difference is:

$f(x) - g(x) = -102x^3 + 56x^2 + x - 55$

Final Answer:

$f(x) - g(x) = -102x^3 + 56x^2 + x - 55$

Would you like more details on any step or further clarification?

Here are some related questions:

1. How do we factor polynomials like $-102x^3 + 56x^2 + x - 55$?
2. Can we graph $f(x)$ and $g(x)$ to visualize their difference?
3. What is the degree and leading coefficient of the result?
4. How can we check the correctness of the subtraction?
5. Can we solve $f(x) - g(x) = 0$ for specific values of $x$?

Tip: When subtracting functions, ensure that you distribute the negative sign carefully across all terms in the second function.