1. **State the problem:** Simplify or analyze the expression $x^2 + 3y^2 - z^2 + 2yz - 4xy$. 2. **Identify the terms:** The expression contains quadratic terms in $x$, $y$, and $z$, and mixed terms involving products of variables. 3. **Group terms:** Group the expression by variables: $$x^2 - 4xy + 3y^2 + 2yz - z^2$$ 4. **Focus on the $x$ and $y$ terms:** Consider $x^2 - 4xy + 3y^2$. This can be factored as: $$x^2 - 4xy + 3y^2 = (x - 3y)(x - y)$$ 5. **Rewrite the expression:** $$ (x - 3y)(x - y) + 2yz - z^2 $$ 6. **Analyze the remaining terms:** $2yz - z^2$ can be rewritten as: $$ z(2y - z) $$ 7. **Final expression:** $$ (x - 3y)(x - y) + z(2y - z) $$ This is a factored form showing the structure of the original expression. **Answer:** The expression factors as $$ (x - 3y)(x - y) + z(2y - z) $$.