Question

Simplify: ${(ab+bc)}^2–2a{b}^{2}c$

Answer 1

Given expression is ${(ab+bc)}^2–2a{b}^{2}c$:

Using the identities

${(a+b)}^2=a^2+b^2+2ab$

${(a-b)}^2=a^2+b^2-2ab$

In the given expression consider ${(ab+bc)}^2$:

$\Rightarrow$${(ab+bc)}^2$ $=$ ${\left(ab\right)}^2+{\left(bc\right)}^2+2\times ab\times bc$ $\left[\because {(a+b)}^2=a^2+b^2+2ab\right]$

$\Rightarrow$${(ab+bc)}^2$ $=$ $a^2{b}^2+b^2{c}^2+2a{b}^{2}c$

Now, ${(ab+bc)}^2–2a{b}^{2}c$$=$$a^2{b}^2+b^2{c}^2+2a{b}^{2}c$$–2a{b}^{2}c$

Therefore, ${(ab+bc)}^2–2a{b}^{2}c$$=$$a^2{b}^2+b^2{c}^2$.