f. \( 3 x\left(4 p^{2}+x^{3}\right) \)

Let's talk about factoring polynomials! The expression \( 3x(4p^2 + x^3) \) is already in a simplified factored form, where \( 3x \) is a common factor. When working with polynomials, look for common factors, then apply techniques like grouping or the distributive property. For instance, if you were to expand it back out, you would multiply \( 3x \) by each term inside the parentheses, leading to a clearer picture of its structure. Another fun concept is the roots of polynomials. If you ever set \( 3x(4p^2 + x^3) = 0 \) to find the values of \( x \) and \( p \), you can see that one root is simply \( x = 0 \) (easy peasy!). For the second part, you would set \( 4p^2 + x^3 = 0 \). Solving that opens up a world of equations where you can play with values and relationships between \( p \) and \( x \)!