Expand the following by using suitable identities, a+2b+5c2 .

Expand the following by using suitable identities, ${\left(a+2b+5c\right)}^{2}$ .

1. A
${a}^{2}-4{b}^{2}+25{c}^{2}+4\mathit{ab}+20\mathit{bc}+10\mathit{ca}$
2. B
${a}^{2}+4{b}^{2}-25{c}^{2}+4\mathit{ab}+20\mathit{bc}+10\mathit{ca}$
3. C
${a}^{2}+4{b}^{2}+25{c}^{2}-4\mathit{ab}+20\mathit{bc}+10\mathit{ca}$
4. D
${a}^{2}+4{b}^{2}+25{c}^{2}+4\mathit{ab}+20\mathit{bc}+10\mathit{ca}$

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Solution:

It is given ${\left(a+2b+5c\right)}^{2}$.
We know the algebraic identity,
${\left(x+y+z\right)}^{2}={x}^{2}+{y}^{2}+{z}^{2}+2\mathit{xy}+2\mathit{yz}+2\mathit{xz}$.
Applying algebraic identity in ${\left(a+2b+5c\right)}^{2}$, where , we get,
${\left(a+2b+5c\right)}^{2}={a}^{2}+{\left(2b\right)}^{2}+{\left(5c\right)}^{2}+2\left(a\right)\left(2b\right)+2\left(2b\right)\left(5c\right)+2\left(5c\right)\left(a\right)$
${\left(a+2b+5c\right)}^{2}={a}^{2}+4{b}^{2}+25{c}^{2}+2\left(2\mathit{ab}\right)+2\left(10\mathit{bc}\right)+2\left(5\mathit{ca}\right)$
${\left(a+2b+5c\right)}^{2}={a}^{2}+4{b}^{2}+25{c}^{2}+4\mathit{ab}+20\mathit{bc}+10\mathit{ca}$
Therefore, option 4 is  correct.

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