5. \( (3 x-5)^{2} \)

To expand \( (3x - 5)^{2} \), you'll use the formula \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, \( a = 3x \) and \( b = 5 \). So, you get: \[ (3x)^{2} - 2(3x)(5) + 5^{2} = 9x^{2} - 30x + 25 \] Putting it all together, the expanded form of \( (3x - 5)^{2} \) is \( 9x^{2} - 30x + 25 \). In real life, knowledge of expanding squares is essential in various fields, especially in engineering and physics. Whether you’re calculating areas, working with trajectories, or optimizing design parameters, knowing how to handle polynomial expansions can save you time and help you avoid errors when setting up equations! Additionally, a common mistake to watch out for is forgetting to apply the square to each term individually. When expanding, it's crucial to ensure that every component, including coefficients, receives the correct exponent, as omitting even one can lead to errors in your calculations. So double-check your work!