Question

Find the square root of $\left[\right(x^2+5x+6\left)\right(x^2+7x+10\left)\right(x^2+8x+15\left)\right]$

Answer 1

Given $\sqrt{\left[\right(x^2+5x+6\left)\right(x^2+7x+10\left)\right(x^2+8x+15\left)\right]}$
Lets factorize, each polynomial.
Consider $x^2+5x+6=x^2+3x+2x+6$ [Since, $5x=3x+2x$]
$=x(x+3)+2(x+3)$
$\therefore x^2+5x+6=(x+3)(x+2)$
Similar way, consider $(x^2+7x+10)=x^2+5x+2x+10$
[Since, $7x=5x+2x$]
$=x(x+5)+2(x+5)$
$\therefore (x^2+7x+10)=(x+5)(x+2)$
Now, consider $(x^2+8x+15)=x^2+5x+3x+15$
$=x(x+5)+3(x+5)$
$\therefore (x^2+8x+15)=(x+5)(x+3)$
Thus, given expression can be written as $(x^2+5x+6)(x^2+7x+10)(x^2+8x+15)=(x+3)(x+2)(x+5)(x+2)(x+5)(x+3)$
$=(x+2{)}^2(x+3{)}^2(x+5{)}^2$
$\Rightarrow (x^2+5x+6)(x^2+7x+10)(x^2+8x+15)=\left[\right(x+2\left)\right(x+3\left)\right(x+5){]}^2$
Taking square root on both sides, we get
$\Rightarrow \sqrt{(x^2+5x+6)(x^2+7x+10)(x^2+8x+15)}=\sqrt{\left[\right(x+2\left)\right(x+3\left)\right(x+5){]}^2}$
$\therefore \sqrt{(x^2+5x+6)(x^2+7x+10)(x^2+8x+15)}=(x+2)(x+3)(x+5)$