Suppose that $f(x+h)-f(x)=-6 h x-8 h+3 h^{2}$. Find: \[ f^{\prime}(x)= \]

Step 1 ：Suppose that \(f(x+h)-f(x)=-6 h x-8 h+3 h^{2}\). Find \(f^{prime}(x)\) by finding the limit as h approaches 0 of the difference quotient:

Step 2 ：\[\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Step 3 ：Plug in the expression for \(f(x+h) - f(x)\) and simplify:

Step 4 ：\[\lim_{h \to 0} \frac{3h^{2} - 6hx - 8h}{h}\]

Step 5 ：Factor out h from the numerator:

Step 6 ：\[\lim_{h \to 0} \frac{h(3h - 6x - 8)}{h}\]

Step 7 ：Cancel out h from the numerator and denominator:

Step 8 ：\[\lim_{h \to 0} (3h - 6x - 8)\]

Step 9 ：Take the limit as h approaches 0:

Step 10 ：\[3(0) - 6x - 8\]

Step 11 ：Final Answer: \(\boxed{f'(x) = -6x - 8}\)