Question

100 points PLEASE ANSWER ASAP thank you so much

Answer 1

LHS

• (a+bi)²
• (a+bi)(a+bi)
• a²+b²i²+2abi
• a²-b²+2abi

RHS

• (a+b)(a-b)+2abi
• a²-b²+2abi

Hence verified

Answer 2

`\textsf{Apply the Perfect Square Formula}\quad (a + b)^2 = a^2 + 2ab + b^2:`

`\implies (a+bi)^2=a^2+2abi+(bi)^2`

`\textsf{Apply exponent rule} \quad (a\cdot b)^c=a^cb^c:`

`\implies a^2+2abi+b^2i^2`

`i=√(-1) \implies i^2=-1`

`\textsf{Apply imaginary number rule} \quad i^2=-1:`

`\implies a^2+2abi+b^2(-1)`

`\implies a^2+2abi-b^2`

`\textsf{Collect the real parts:}`

`\implies a^2-b^2+2abi`

`\textsf{Factor the real parts by applying the Difference of Two Squares Formula}\\x^2-y^2=(x+y)(x-y) :`

`\implies (a+b)(a-b)+2abi`