## Intermediate Algebra: Connecting Concepts through Application

$(2x-3)(4x^2+6x+9)$
The student incorrectly used the factoring of the difference of $2$ squares. The factoring of the difference of $2$ cubes should instead be used. The expressions $8x^3$ and $27$ are both perfect cubes (the cube root is exact). Hence, $8x^3-27$ is a $\text{ difference }$ of $2$ cubes. Using the factoring of the sum or difference of $2$ cubes, which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$, the expression, $8x^3-27,$ is equivalent to \begin{array}{l}\require{cancel} (2x)^3-(3)^3 \\\\= (2x-3)[(2x)^2+2x(3)+(3)^2] \\\\= (2x-3)(4x^2+6x+9) .\end{array}