Solve for the difference (x)= -4x(12x^2-5x-11)  and g(x)=54x^3-36x^2+43x+55

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.

Solve for the difference f(x)= -4x(12x^2-5x-11) and g(x)=54x^3-36x^2+43x+55

Learn how to subtract two polynomial functions with a step-by-step solution. This problem involves expanding f(x), using the distributive property, and simplifying the result by combining like terms. Suitable for Grades 10-12 students studying polynomial functions.

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