Factoring out a monomial from a polynomial: Multivariate Factor the following expression. \[ 30 u^{4} v^{7}-6 u^{7} v^{6} y^{3} \]

Step 1 ：Identify the greatest common factor (GCF) of all the terms in the polynomial. In this case, the GCF is the monomial that appears in each term with the smallest exponent. Looking at the given expression, we can see that the terms share a common factor of \(6u^{4}v^{6}\).

Step 2 ：Factor out this monomial from each term in the polynomial to simplify the expression.

Step 3 ：The factored form of the polynomial is \(-6u^{4}v^{6}(u^{3}y^{3} - 5v)\). This is the simplest form of the given polynomial.

Step 4 ：Final Answer: The factored form of the polynomial \(30 u^{4} v^{7}-6 u^{7} v^{6} y^{3}\) is \(\boxed{-6u^{4}v^{6}(u^{3}y^{3} - 5v)}\).