Remember that $$ f(x)=5 x-3 x^{2} $$ Find $$ f(x+h)-f(x) $$ $$ \square $$

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To find $$ f(x+h) - f(x) $$ for the function $$ f(x) = 5x - 3x^2 $$, we first need to calculate $$ f(x+h) $$.

1. **Calculate $$ f(x+h) $$**:
   $$
   f(x+h) = 5(x+h) - 3(x+h)^2
   $$
   Expanding this:
   $$
   f(x+h) = 5x + 5h - 3((x+h)^2)
   $$
   Now, expand $$ (x+h)^2 $$:
   $$
   (x+h)^2 = x^2 + 2xh + h^2
   $$
   Therefore,
   $$
   f(x+h) = 5x + 5h - 3(x^2 + 2xh + h^2)
   $$
   Distributing the $$-3$$:
   $$
   f(x+h) = 5x + 5h - 3x^2 - 6xh - 3h^2
   $$
   Combining like terms:
   $$
   f(x+h) = (5x - 3x^2) + (5h - 6xh - 3h^2)
   $$
   Thus,
   $$
   f(x+h) = f(x) + (5h - 6xh - 3h^2)
   $$

2. **Now, calculate $$ f(x+h) - f(x) $$**:
   $$
   f(x+h) - f(x) = (f(x) + (5h - 6xh - 3h^2)) - f(x)
   $$
   This simplifies to:
   $$
   f(x+h) - f(x) = 5h - 6xh - 3h^2
   $$

Therefore, the final result is:
$$
\boxed{5h - 6xh - 3h^2}
$$
$$ f(x+h) - f(x) = 5h - 6xh - 3h^2 $$

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