How do you factor completely $3{x^3} + 24{x^2} + 48x$?

Hint: First we will reduce the equation further if possible. Then we will try to factorize the terms in the equation. Then solve the equation by using the expansion of ${a^2} - {b^2}$ and find the appropriate solution for the equation. Split the middle term and finally factorize the final term.

Complete step by step answer:
We will start off by reducing any reducible terms in the equation.
$\Rightarrow$ $3{x^3} + 24{x^2} + 48x$
$\Rightarrow$ Now we will factorize the terms in the equation.
$
  3{x^3} + 24{x^2} + 48x \\
  3{x^3} + 8 \times 3{x^2} + 48x \\
  3{x^3} + {2^3} \times 3{x^2} + 48x \\
 $
$\Rightarrow$ Now we will pull out the like terms from the equation.
$
   = 3{x^3} + 24{x^2} + 48x \\
   = 3x \times ({x^2} + 8x + 16) \\
 $
$\Rightarrow$ Now we will try to factorize by splitting the middle term.
$
   = 3x \times ({x^2} + 8x + 16) \\
   = 3x \times ({x^2} + 4x + 4x + 16) \\
   = 3x \times (x(x + 4) + 4(x + 4)) \\
   = 3x \times (x + 4)(x + 4) \\
   = 3x \times {(x + 4)^2} \\
 $

Note:
 While splitting the middle term be careful. After splitting the middle term Do not solve all the equations simultaneously. Solve all the equations separately, so that you don’t miss any term of the solution. Check if the solution satisfies the original equation completely. If any term of the solution doesn’t satisfy the equation, then that term will not be considered as a part of the solution.