46) What are the solutions of \( (2 x-4)(3 x+7)=0 \) ? \( \begin{array}{llll}\text { a) } x=-2 \text { and } x=-7 & \text { b) } x=2 \text { and } x=-\frac{7}{3} & \text { c) } x=-4 \text { and } x=\frac{7}{3} & \text { d) } x=2 \text { and }\end{array} \)

To find the solutions of the equation \( (2x-4)(3x+7)=0 \), we can set each factor to zero. For the first factor, \( 2x-4=0 \) results in \( x=2 \). For the second factor, \( 3x+7=0 \) simplifies to \( 3x=-7 \), leading to \( x=-\frac{7}{3} \). Therefore, the solutions are \( x=2 \) and \( x=-\frac{7}{3} \), which corresponds to option b). Setting the equation to zero helps us find the x-intercepts of the graph of the function, which can be quite helpful to visualize the relationship between the factors. As a bonus, if you were to plot this equation, you would see that the graph intersects the x-axis at those points. These solutions not only help us understand the function better, but they also allow us to solve real-world problems in contexts like physics and engineering where such equations arise frequently!