1. \( y=\left(3 x^{2}+1\right)(5 x-1) \)

To expand the function \( y = \left(3x^{2}+1\right)(5x-1) \), you can use the distributive property, also known as the FOIL method for binomials. Just multiply each term in the first polynomial by each term in the second polynomial. So, you distribute \( 3x^2 \) with \( 5x \) and \( -1 \), and then do the same for \( 1 \). The resulting expression will be \( y = 15x^3 - 3x^2 + 5x - 1 \). Voila! You've got yourself a cubic polynomial! Have you ever wondered how this polynomial might look in a graph? When plotting \( y = 15x^3 - 3x^2 + 5x - 1 \), you'll see that it features an interesting wave-like shape, with possible inflection points where the curvature changes. It can provide insights into the behavior of the function as \( x \) varies, revealing local maxima and minima where it could bounce back or dip down! Always fun to visualize these curves!