Step 1

Multiply A with the terms in the first bracket:

\(\displaystyle={\left[{A}{\left({A}'+{B}'\right)}\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)

\(\displaystyle={\left[\forall'+{A}{B}'\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)

Since \(PP'=0\)

\(\displaystyle={\left[{0}+{A}{B}'\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)

\(\displaystyle={A}{B}'{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}\)

Step 2

Multiply AB' with the terms in the first bracket of the obtained expression:

\(\displaystyle={\left[{A}{B}'{\left({B}+{C}\right)}\right]}{\left({B}+{C}'+{D}\right)}\)

\(\displaystyle={\left[{A}{B}{B}'+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}\)

\(\displaystyle={\left[{A}{\left({B}{B}'\right)}+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}\)

Since \(PP'=0\)

\(\displaystyle={\left[{0}+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}\)

\(\displaystyle={A}{B}'{C}{\left({B}+{C}'+{D}\right)}\)

Step 3

Multiply AB'C with all the terms in the bracket of the obtained expression:

\(=AB'C(B+C'+D)\)

\(=ABB'C+AB'CC'+AB'CD\)

\(=A(BB')C+AB'(CC')+AB'CD\)

Since \(PP'=0\)

\(=0+0+AB'CD\)

\(=AB'CD\)

Thus the simplified expression is AB'CD.