## Intermediate Algebra (6th Edition)

$(x^3-6)(x^3+2)$
Let $z=x^3$. Then the given expression, $x^6-4x^3-12$, is equivalent to $z^2-4z-12$. The two numbers whose product is $ac= 1(-12)=-12$ and whose sum is $b= -4$ are $\{ -6,2 \}$. Using these two numbers to decompose the middle term, then the factored form of the expression, $z^2-4z-12$, is \begin{array}{l}\require{cancel} z^2-6z+2z-12 \\\\= (z^2-6z)+(2z-12) \\\\= z(z-6)+2(z-6) \\\\= (z-6)(z+2) .\end{array} Since $z=x^3$, then, \begin{array}{l} (z-6)(z+2) \\\\= (x^3-6)(x^3+2) .\end{array}