Question

factorize:
${x}^{2} + {y}^{2} + x + y + 2xy$
​

Answer 1

Here,

$\mapsto \: \bf{x}^{2} + {y}^{2} + x + y + 2xy$

• Rearranging the terms,

$\to \bf{x}^{2} + {y}^{2}+ 2xy + x + y$

• Taking certain terms under brackets

$\to\bf({x}^{2} + {y}^{2}+ 2xy) + (x + y )$

• Factoring the terms in bracket (only if possible)
• In above expression, ${x}^{2} + {y}^{2}+ 2xy$ can be factorised using --

$\small \text{ \green{Identity } \gray : } \pink{\rm {a}^{2} + {b}^{2} + 2ab = {(a + b)}^{2} }$

Therefore,

$\to\bf[ {(x + y)}^{2} ]+ (x + y)$

• Taking (x + y) as common,

$\to \bf{}x + y[(x + y) + (1)]$

• Further solving in brackets,

$\to\bf \: \red{(x + y)(x + y + 1)}$

Hence,

• On factorizing ${x}^{2} + {y}^{2} + x + y + 2xy$, $\sf \: \red{(x + y)(x + y + 1)}$ is obtained.