Factor completely. 20p^3+30p^2+18p+27 square

To factor the polynomial \(20p^3 + 30p^2 + 18p + 27\) completely, we can follow these steps:<br /><br />1. **Group the terms**: Group the polynomial into two pairs.<br /> \[<br /> (20p^3 + 30p^2) + (18p + 27)<br /> \]<br /><br />2. **Factor out the greatest common factor (GCF) from each group**:<br /> \[<br /> 10p^2(2p + 3) + 9(2p + 3)<br /> \]<br /><br />3. **Factor by grouping**: Notice that \((2p + 3)\) is a common factor.<br /> \[<br /> (10p^2 + 9)(2p + 3)<br /> \]<br /><br />4. **Check for further factoring**: The quadratic \(10p^2 + 9\) does not factor further over the integers.<br /><br />Thus, the completely factored form of the polynomial \(20p^3 + 30p^2 + 18p + 27\) is:<br />\[<br />(10p^2 + 9)(2p + 3)<br />\]