Is it possible to find irreducible polynomial of this form..

Question

I am searching for a sextic polynomial $f(x)$ that is irreducible over $\mathbb Q$ and factors in the following way over $\mathbb R$

$f(x)=((x+a)^2+d^2)((x+b)^2+d^2)((x+c)^2+d^2)$

where $d$ is a non-zero real number and $a,b,c$ are real numbers that are mutually distinct. Is this even possible?

Edit: Will Jagy just gave an example that is worth marking as answer. So now that we know that such polynomials over $\mathbb Q$ exists. Could we also prove that the splitting field of these polynomials, say $K$, will have a Galois group $Gal(K/\mathbb Q)$ that is never symmetric (or maybe could we even say that it is always cyclic?)

Answer 1

$$ x^6 - 3 x^4 + 2 x^3 + 12 x^2 - 12 x + 17 $$

You used + signs with a,b,c, so it comes out a little peculiar compared with how I built it: $$ a = 2 \cos \frac{7 \pi}{9} \; , \; \; b = 2 \cos \frac{5 \pi}{9} \; , \; \; c = 2 \cos \frac{ \pi}{9} \; , \; \; d=1 \; \; . $$

I started with the famous $z^3 - 3 z + 1.$ Because of the sign change, $a,b,c$ above are the roots of $w^3 - 3 w - 1 \; .$

Answer 2

Begin with $x^3 - 6x + 1,$ three real irrational roots $a,b,c, \;$ Galois group symmetric (discriminant is $837= 27 \cdot 31$). Calculate $$ \left( (x-a)^2 + 1 \right)\left( (x-b)^2 + 1 \right)\left( (x-c)^2 + 1 \right) $$ or $$ x^6 - 9 x^4 + 2 x^3 + 39 x^2 - 18 x + 50 $$ $$ $$ $$ $$

parisize = 4000000, primelimit = 500000
?  polgalois( x^3 - x + 1)
%1 = [6, -1, 1, "S3"]
? pp = polroots( x^3 - x + 1)
%2 = [-1.324717957244746025960908855 + 0.E-28*I, 0.6623589786223730129804544272 - 0.5622795120623012438991821449*I, 0.6623589786223730129804544272 + 0.5622795120623012438991821449*I]~
?  polgalois( x^3 -3 * x + 1)
%3 = [3, 1, 1, "A3"]





?  polgalois( x^3 -6 * x + 1)
%4 = [6, -1, 1, "S3"]
? pp = polroots( x^3 -6 * x + 1)
%5 = [-2.528917957294361733726484439 + 0.E-28*I, 0.1674491911085351562744105995 + 0.E-28*I, 2.361468766185826577452073839 + 0.E-28*I]~
? pp1 = pp[1]
%6 = -2.528917957294361733726484439 + 0.E-28*I
? pp2 = pp[2]
%7 = 0.1674491911085351562744105995 + 0.E-28*I
? pp3 = pp[3]
%8 = 2.361468766185826577452073839 + 0.E-28*I
? x^2 - 2 * pp1 * x + pp1^2 + 1
%9 = x^2 + (5.057835914588723467452968878 + 0.E-28*I)*x + (7.395426034725887197635766878 + 0.E-28*I)


? r = (x^2 - 2 * pp1 * x + pp1^2 + 1)  *  (x^2 - 2 * pp2 * x + pp2^2 + 1)  * (x^2 - 2 * pp3 * x + pp3^2 + 1)  
%10 = x^6 + (1.009741959 E-28 + 0.E-28*I)*x^5 + (-9.000000000000000000000000001 + 0.E-27*I)*x^4 + (2.000000000000000000000000001 + 0.E-27*I)*x^3 + (39.00000000000000000000000000 + 0.E-27*I)*x^2 + (-18.00000000000000000000000000 + 0.E-27*I)*x + (50.00000000000000000000000000 + 0.E-27*I)
? 
? g = x^6 - 9 * x^4 + 2 * x^3 + 39 * x^2 - 18 * x + 50
%11 = x^6 - 9*x^4 + 2*x^3 + 39*x^2 - 18*x + 50
? factor(g)
%12 = 
[x^6 - 9*x^4 + 2*x^3 + 39*x^2 - 18*x + 50 1]

? polroots(g)
%13 = [-2.528917957294361733726484439  - 1.000000000000000000000000000*I,
       -2.528917957294361733726484439  + 1.000000000000000000000000000*I,
        0.1674491911085351562744105995 - 1.000000000000000000000000000*I, 
        0.1674491911085351562744105995 + 1.000000000000000000000000000*I,
        2.361468766185826577452073839  - 1.000000000000000000000000000*I,
        2.361468766185826577452073839  + 1.000000000000000000000000000*I]~
? 

groups of the sextics

? g
%19 = x^6 - 9*x^4 + 2*x^3 + 39*x^2 - 18*x + 50
? polgalois(g)
%20 = [12, -1, 1, "D(6) = S(3)[x]2"]
?  rr = (x^2 - 2 * pp1 * x + pp1^2 + 2)  *  (x^2 - 2 * pp2 * x + pp2^2 + 2)  * (x^2 - 2 * pp3 * x + pp3^2 + 2)
%21 = x^6 + (1.009741959 E-28 + 0.E-28*I)*x^5 + (-6.000000000000000000000000001 + 0.E-27*I)*x^4 + (2.000000000000000000000000001 + 0.E-27*I)*x^3 + (48.00000000000000000000000000 + 0.E-27*I)*x^2 + (-24.00000000000000000000000000 + 0.E-27*I)*x + (129.0000000000000000000000000 + 0.E-27*I)
? gg = x^6 -6 * x^4 + 2 * x^3 + 48 * x^2 -24 * x + 129
%22 = x^6 - 6*x^4 + 2*x^3 + 48*x^2 - 24*x + 129
? polroots(gg)
%23 = [-2.528917957294361733726484439 - 1.414213562373095048801688724*I, -2.528917957294361733726484439 + 1.414213562373095048801688724*I, 0.1674491911085351562744105995 - 1.414213562373095048801688724*I, 0.1674491911085351562744105995 + 1.414213562373095048801688724*I, 2.361468766185826577452073839 - 1.414213562373095048801688724*I, 2.361468766185826577452073839 + 1.414213562373095048801688724*I]~
? polgalois(gg)
%24 = [12, -1, 1, "D(6) = S(3)[x]2"]
?