Answer
$$3x^2(x - 2)(x^2 + 2x + 4)$$
Work Step by Step
To factor this polynomial, we need to factor out the greatest common factor of both the coefficient and the variable:
The greatest common factor of $3$ and $24$ is $3$.
The greatest common factor of $x^5$ and $x^2$ is $x^2$.
Therefore, we will factor out $3x^2$ from each term to get:
$$3x^2(x^3 - 8)$$
We see that we now have a polynomial that is the difference of two cubes, so we can further factor this polynomial. The formula for factoring the difference of two cubes is given as:
$$A^3 - B^3 = (A - B)(A^2 + AB + B^2)$$
We now plug in the values for $A$, which is the $\sqrt[3] x$ or $x$, and $B$, which is the $\sqrt[3] 8$ or $2$, to get:
$$3x^2(x - 2)(x^2 + 2x + 2^2)$$
Multiply to simplify:
$$3x^2(x - 2)(x^2 + 2x + 4)$$
