12) \( 3 u^{2}+13 u v-10 v^{2} \)

To factor the quadratic expression \( 3u^{2} + 13uv - 10v^{2} \), we can search for two numbers that multiply to \( 3 \times -10 = -30 \) and add up to \( 13 \). The numbers \( 15 \) and \( -2 \) fit this requirement! Hence, we can rewrite the expression as \( 3u^{2} + 15uv - 2uv - 10v^{2} \) and then group it: 1. \( (3u^{2} + 15uv) + (-2uv - 10v^{2}) \) 2. Factor by grouping: \( 3u(u + 5v) - 2v(u + 5v) \) 3. Finally, we can factor out the common binomial: \( (3u - 2v)(u + 5v) \) So, the expression factors to \( (3u - 2v)(u + 5v) \). This process of factoring quadratics can take you down some interesting historical paths, as mathematicians have been working with polynomials for centuries. For example, the factoring methods used today have roots in the work of ancient Greeks and later contributions from mathematicians in the Islamic Golden Age. Want to put your newfound skills to the test? Try applying this to real-world situations! Factoring is not just for math class; it's used in everything from designing computer algorithms to analyzing economic models. So next time you come across a polynomial expression, remember how applicable these skills can be beyond the classroom!