We can check that the equation is true by expanding the lhs and rhs to see if they are the same. Here’s the lhs: \begin{align*} (a+b)^3 &= (a+b)(a+b)^2\\ &= (a+b)(a^2 + 2ab + b^2)\\ &= a^3 + 2a^2b + ab^2 + ba^2 + 2ab^2 + b^3\\ &= a^3 + 3a^2b + 3ab^2 + b^3. \end{align*} And the rhs: \begin{align*} 3ab(a+b) + (a^3 + b^3) &= 3aba + 3abb + a^3 + b^3\\ &= 3a^2b + 3ab^2 + a^3 + b^3\\ &= a^3 + 3a^2b + 3ab^2 + b^3, \end{align*} which is the same as the lhs, so the equation is true.

Or you might choose to manipulate the lhs until it becomes the rhs: \begin{align*} (a+b)^3 &= a^3 + 3a^2b + 3ab^2 + b^3\\ &= 3a^2b + 3ab^2 + (a^3 + b^3)\\ &= 3ab(a + b) + (a^3 + b^3). \end{align*}