To find $f'(x)$, we will use the definition of the derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$

Given that $f(x+h) - f(x) = -5hx^2 - 8hx + 6h^2x - 3h^2 + 7h^3$, we can plug this into the formula:

$f'(x) = \lim_{h \to 0} \frac{-5hx^2 - 8hx + 6h^2x - 3h^2 + 7h^3}{h}.$

Now, simplify the expression by dividing each term by $h$:

$f'(x) = \lim_{h \to 0} (-5x^2 - 8x + 6hx - 3h + 7h^2).$

As $h \to 0$, the terms involving $h$, $h^2$, and $h^3$ vanish, leaving us with:

$f'(x) = -5x^2 - 8x.$

So, the derivative is:

$f'(x) = -5x^2 - 8x.$

Would you like further details or have any questions? Here are five related questions to explore:

1. What is the general process of finding a derivative using the limit definition?
2. How do higher powers of $h$ disappear as $h \to 0$?
3. Can you find the second derivative $f''(x)$ from the expression we derived?
4. How does this method compare to using basic differentiation rules (like the power rule)?
5. How does the behavior of $f'(x)$ change as $x$ increases or decreases?

Tip: When simplifying a derivative expression, always carefully consider which terms will vanish as $h \to 0$ to avoid unnecessary complexity.