Find the square root of $(x^2 + 3x + 7)(x^2 + 5x + 3) + (x − 2)^2$

Question

I want to find the square root of $$(x^2+3x + 7)(x^2+5x+3)+ (x −2)^2$$
First , I would like to know if it is really necessary to expand everything , because I think it is in the given form for a special reason.

Anyway I expanded and got $$x^4+8 x^3+26 x^2+40 x+25$$ and after that also rational roots test is of no use because there are no real factors.

Please help.

Answer 1

Given that you have already expanded, you might as well equate coefficients of $(x^2 + ax + 5)^2$ as has already been commented.

Perhaps the reason why it is in the form given is, let us say the square root is $p(x)$.

Then, $(x^2+3x + 7)(x^2+5x+3) = p^2 - (x-2)^2 = (p - x + 2) (p + x - 2)$

Now can you read off what $p(x)$ could be?
(Addnl hint, $p$ is just the average of the factors)

Answer 2

As $x^2+3x + 7-(x^2+5x+3)=-2(x-2), x^2+5x+3=(x^2+3x +7)+2(x-2) $

$\implies (x^2+3x +7)(x^2+5x+3)+(x-2)^2$

$=(x^2+3x +7)\{x^2+3x +7+2(x-2)\}+(x-2)^2$

$=(x^2+3x +7)^2+2(x^2+3x +7)(x-2)+(x-2)^2$

$=\{(x^2+3x +7)+(x-2)\}^2$

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Alternatively, if we $x^2+3x + 7=a, x^2+5x+3=b, x-2=c\implies b-a=2c$

$\implies (x^2+3x +7)(x^2+5x+3)+(x-2)^2=a\cdot b +c^2$ $=a(a+2c)+c^2=a^2+c^2+2ca=(a+c)^2$