To simplify the expression
\[ \frac{15s^4 + 9s^3 - 21s^2 - 3s}{-3s} \]
we can factor the numerator.
First, we can factor out a common term from the numerator:
\[ 15s^4 + 9s^3 - 21s^2 - 3s \]
Notice that each term has \(s\) as a common factor:
\[ = s(15s^3 + 9s^2 - 21s - 3) \]
Next, we can factor \(15s^3 + 9s^2 - 21s - 3\). Pair the terms for easy factoring:
\[ = s((15s^3 + 9s^2) + (-21s - 3)) \]
Factor out the common factors in each pair:
\[ = s(3s^2(5s + 3) - 3(7s + 1)) \] Now we continue to factor by observing that both groups contain the factor \(3\):
Let's check if we can further factor \(5s + 3\) and \(7s + 1\):
So, now we have:
\[ = s(3(5s + 3)(s - 1)) \]
Therefore, the entire expression can be rewritten as:
\[ = \frac{s(3(5s + 3)(s - 1))}{-3s} \]
Now we can cancel \(s\) in the numerator and denominator (assuming \(s \neq 0\)):
\[ = \frac{3(5s + 3)(s - 1)}{-3} \]
The \(3\)s will cancel out:
\[ = -(5s + 3)(s - 1) \]
Now, expanding this out gives us:
\[ = -[(5s^2 - 5s + 3s - 3)] \] \[ = -[5s^2 - 5s + 3s - 3] \] \[ = -5s^2 + 2s + 3 \]
Thus, the simplified expression is:
\[ \boxed{-5s^2 + 2s + 3} \]
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