\( \leftarrow \quad \begin{array}{l}\text { Factor. Check by multiplying. } \\ 6 x^{6}+6 x^{5}-18 x^{4}+30 x^{3}\end{array} \)

To factor the polynomial \( 6x^{6}+6x^{5}-18x^{4}+30x^{3} \), we can start by factoring out the greatest common factor (GCF), which here is \( 6x^{3} \). This gives us: \[ 6x^{3}(x^{3} + x^{2} - 3x + 5) \] Now, let’s check our work by distributing \( 6x^{3} \) back into the polynomial: \[ 6x^{3} \cdot x^{3} + 6x^{3} \cdot x^{2} - 6x^{3} \cdot 3x + 6x^{3} \cdot 5 = 6x^{6} + 6x^{5} - 18x^{4} + 30x^{3} \] Since we get back our original polynomial, the factorization is confirmed as correct! To go a step further, we can also explore synthetic division or factoring by grouping to check for any further simplifications. However, the cubic \( x^{3} + x^{2} - 3x + 5 \) does not factor neatly with rational roots, so this factorization is likely as simplified as we can get with integers. Keep practicing, and you'll be a factoring pro in no time!