Factor the expression completely. -9+45x^3 square

To factor the expression \(-9 + 45x^3\) completely, we first look for the greatest common factor (GCF) of the terms.<br /><br />The terms are \(-9\) and \(45x^3\). The GCF of \(-9\) and \(45\) is \(9\), and since the first term is negative, we factor out \(-9\):<br /><br />\[<br />-9 + 45x^3 = -9(1 - 5x^3)<br />\]<br /><br />Next, we need to factor \(1 - 5x^3\). This is a difference of cubes, which can be factored using the formula:<br /><br />\[<br />a^3 - b^3 = (a - b)(a^2 + ab + b^2)<br />\]<br /><br />In this case, \(a = 1\) and \(b = \sqrt[3]{5x^3} = \sqrt[3]{5}x\). Applying the formula:<br /><br />\[<br />1 - 5x^3 = (1 - \sqrt[3]{5}x)(1^2 + 1 \cdot \sqrt[3]{5}x + (\sqrt[3]{5}x)^2)<br />\]<br /><br />Simplifying the terms inside the parentheses:<br /><br />\[<br />1 - 5x^3 = (1 - \sqrt[3]{5}x)(1 + \sqrt[3]{5}x + (\sqrt[3]{5}x)^2)<br />\]<br /><br />Thus, the completely factored form of the expression \(-9 + 45x^3\) is:<br /><br />\[<br />-9(1 - 5x^3) = -9(1 - \sqrt[3]{5}x)(1 + \sqrt[3]{5}x + (\sqrt[3]{5}x)^2)<br />\]<br /><br />So, the final answer is:<br /><br />\[<br />\boxed{-9(1 - \sqrt[3]{5}x)(1 + \sqrt[3]{5}x + (\sqrt[3]{5}x)^2)}<br />\]