We start by removing the brackets \((a+b)^3\) \[ \begin{array}{rcl} (a+b)^3 &=& (a+b)(a+b)^2 \\ &&\phantom{myspace}\color{blue}{x^3=x^2\cdot x} \\ &=& (a+b)(a^2+2ab+b^2)\\&&\phantom{myspace}\color{blue}{\text{formula for }(a+b)^2}\\ &=& a(a^2+2ab+b^2)+b(a^2+2ab+b^2)\\&&\phantom{myspace}\color{blue}{\text{removing a part of the brackets}}\\ &=& a^3+2a^2b+\phantom{2}ab^2\\ &&\phantom{a^3}+\phantom{2}a^2b+2ab^2+b^3 \\&&\phantom{myspace}\color{blue}{\text{removing all the brackets}}\\ &=& a^3+3a^2b+3ab^2+b^3\\&&\phantom{myspace}\color{blue}{\text{taking similar terms together}}\end {array}\]

With the formula for \((a+b)^3\) we can remove the brackets in \((a+b)^4\): \[ \begin{array}{rcl} (a+b)^4 &=& (a+b)(a+b)^3 \\ &&\phantom{myspace}\color{blue}{x^4=x^3\cdot x} \\ &=& (a+b)(a^3+3a^2b+3ab^2+b^3)\\ &&\phantom{myspace}\color{blue}{\text{formula for }(a+b)^3}\\ &=& a(a^3+3a^2b+3ab^2+b^3)+b(a^3+3a^2b+3ab^2+b^3)\\&&\phantom{myspace}\color{blue}{\text{removing a part of the brackets}}\\ &=& a^4+3a^3b+3a^2b^2+\phantom{3}ab^3\\ &&\phantom{} \phantom{a^4} +\phantom{3}a^3b+3a^2b^2+3ab^3+b^4\\&&\phantom{myspace}\color{blue}{\text{removing all the brackets}}\\ &=& a^4+4a^3b+6a^2b^2+ 4ab^3+b^4\\&&\phantom{myspace}\color{blue}{\text{taking similar terms together}}\end {array}\]