Find the LCM of $x^3+8,x^2-4$

We know that $LCM$ is the least common multiple.

Factorise $x^3+8$ as follows:

$x^3+8=x^3+2^3=(x+2)(x^2+2^2-2x)$ ............(using identity $a^3+b^3=(a+b)(a^2+b^2-ab)$)

$=(x+2)(x^2+4-2x)$

Now, factorise $x^2-4$ as follows:

$x^2-4=x^2-2^2=(x+2)(x-2)$ .........(using identity $a^2-b^2=(a+b)(a-b)$)

Therefore, the least common multiple between the polynomials $x^3+8$and $x^2-4$ is:

$(x+2)(x-2)(x^2+4-2x)=(x-2)(x^3+8)$ (using identity $a^3+b^3=(a+b)(a^2+b^2-ab)$)

Hence, the $LCM$ is $(x-2)(x^3+8)$.