Multiply. \[ \frac{x^{2}-4 x y-12 y^{2}}{x^{2}-10 x y+21 y^{2}} \cdot \frac{5 x-15 y}{x+2 y} \] simolity your answer as much as possible.

To simplify the expression \[ \frac{x^{2}-4 x y-12 y^{2}}{x^{2}-10 x y+21 y^{2}} \cdot \frac{5 x-15 y}{x+2 y}, \] we start by factoring the polynomials in the fractions. 1. **Factor the numerators and denominators where possible**: - For \(x^{2} - 4xy - 12y^{2}\): This factors to \((x - 6y)(x + 2y)\). - For \(x^{2} - 10xy + 21y^{2}\): This factors to \((x - 7y)(x - 3y)\). - For \(5x - 15y\): This can be factored out as \(5(x - 3y)\). So, rewriting the expression gives us: \[ \frac{(x - 6y)(x + 2y)}{(x - 7y)(x - 3y)} \cdot \frac{5(x - 3y)}{x + 2y}. \] 2. **Cancel common factors**: Notice that \((x + 2y)\) in the numerator of the first fraction cancels with \((x + 2y)\) in the denominator of the second fraction, and \((x - 3y)\) in the numerator of the second fraction cancels with \((x - 3y)\) in the denominator of the first fraction. This simplifies our expression to: \[ \frac{5(x - 6y)}{x - 7y}. \] 3. **Final Simplification**: Thus, the simplified form of the original expression is \[ \frac{5(x - 6y)}{x - 7y}. \]