We are asked to simplify the following expression: $\frac{-18s^5t(7s+3)}{70st^2(s-2t)} \div \frac{30s^2t^4(3+7s)}{42s^3t^3(2t-s)}$

1. Problem Description

We are asked to simplify the following expression:

$\frac{-18s^5t(7s+3)}{70st^2(s-2t)} \div \frac{30s^2t^4(3+7s)}{42s^3t^3(2t-s)}$

2. Solution Steps

First, recall that dividing by a fraction is the same as multiplying by its reciprocal. Thus, we have:

$\frac{-18s^5t(7s+3)}{70st^2(s-2t)} \div \frac{30s^2t^4(3+7s)}{42s^3t^3(2t-s)} = \frac{-18s^5t(7s+3)}{70st^2(s-2t)} \cdot \frac{42s^3t^3(2t-s)}{30s^2t^4(3+7s)}$

We can rearrange the expression to group like terms together:

$= \frac{-18}{70} \cdot \frac{42}{30} \cdot \frac{s^5}{s} \cdot \frac{s^3}{s^2} \cdot \frac{t}{t^2} \cdot \frac{t^3}{t^4} \cdot \frac{(7s+3)}{(7s+3)} \cdot \frac{(2t-s)}{(s-2t)}$

Simplify the numerical coefficients:

$\frac{-18}{70} \cdot \frac{42}{30} = \frac{-9}{35} \cdot \frac{7}{5} = \frac{-9 \cdot 7}{35 \cdot 5} = \frac{-9 \cdot 7}{5 \cdot 7 \cdot 5} = \frac{-9}{25}$

Simplify the powers of s:

$\frac{s^5}{s} \cdot \frac{s^3}{s^2} = s^{5-1} \cdot s^{3-2} = s^4 \cdot s^1 = s^5$

Simplify the powers of t:

$\frac{t}{t^2} \cdot \frac{t^3}{t^4} = t^{1-2} \cdot t^{3-4} = t^{-1} \cdot t^{-1} = t^{-2} = \frac{1}{t^2}$

Simplify the expressions in parenthesis:

$\frac{(7s+3)}{(7s+3)} = 1$

$\frac{(2t-s)}{(s-2t)} = \frac{-(s-2t)}{(s-2t)} = -1$

Therefore we have:

$\frac{-9}{25} \cdot s^5 \cdot \frac{1}{t^2} \cdot 1 \cdot (-1) = \frac{9s^5}{25t^2}$

3. Final Answer

$\frac{9s^5}{25t^2}$