Finding f(3m-2) for the Function f(x) -5x4-2 - 3

Given the function f(x) -5x4-2 - 3, the goal is to evaluate f(3m-2). This requires a series of substitutions and simplifications. Let's break this down step by step.

Step-by-Step Solution

1. Substitution of 3m-2 for x in the Function

First, let's substitute x 3m-2 into the function f(x) -5x4-2 - 3:

f(3m-2) -5(3m-2)4-2 - 3

2. Simplify the Expression Inside the Parentheses

Next, simplify the expression inside the parentheses:

(3m-2)4-2 (3m-2)2

We now have:

f(3m-2) -5(3m-2)2 - 3

3. Expand (3m-2)2

Expand the square of the binomial (3m-2):

(3m-2)2 (3m-2)(3m-2) 9m2 - 12m 4

Substitute this back into the function:

f(3m-2) -5(9m2 - 12m 4) - 3

4. Distribute -5 Across the Terms

Distribute -5 across the terms inside the parentheses:

-5(9m2 - 12m 4) -45m2 60m - 20

So, the function becomes:

f(3m-2) -45m2 60m - 20 - 3

5. Combine the Constant Terms

Combine the constant terms:

-20 - 3 -23

This results in the final simplified expression for f(3m-2):

f(3m-2) -45m2 60m - 23

Conclusion

Thus, the final result is:

boxed{-45m2 60m - 23}

This walkthrough provides a clear and detailed explanation of how to substitute and simplify the given function f(x) -5x4-2 - 3 with the argument 3m-2. Understanding these steps is crucial for mastering algebraic manipulation and function evaluation.