To solve the equation, we can start by expanding the expression on the left side:

(x^2 - 6x + 18)(x^2 - 10x + 18) + 12x - 39 = 0

(x^4 - 10x^3 + 18x^2 - 6x^3 + 60x^2 - 108x + 18x^2 - 180x + 324) + 12x - 39 = 0

x^4 - 16x^3 + 96x^2 - 288x + 324 + 12x - 39 = 0

x^4 - 16x^3 + 108x^2 - 276x + 285 = 0

Now, we can try to factor the equation. However, it is not easy to find the factors directly. So, we can use the Rational Root Theorem to find possible rational roots.

The Rational Root Theorem states that if a polynomial equation has a rational root p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a possible root.

In this case, the leading coefficient is 1 and the constant term is 285. The factors of 285 are ±1, ±3, ±5, ±15, ±19, ±57, ±95, and ±285. The possible rational roots are therefore:

±1/1, ±3/1, ±5/1, ±15/1, ±19/1, ±57/1, ±95/1, and ±285/1

Simplifying these fractions, we have:

±1, ±3, ±5, ±15, ±19, ±57, ±95, and ±285

We can now try these possible roots one by one to see if they satisfy the equation.

Trying x = 1:

(1^4 - 16(1)^3 + 108(1)^2 - 276(1) + 285) = 1 - 16 + 108 - 276 + 285 = 102

Since 102 is not equal to 0, x = 1 is not a root.

Trying x = -1:

((-1)^4 - 16(-1)^3 + 108(-1)^2 - 276(-1) + 285) = 1 + 16 + 108 + 276 + 285 = 686

Since 686 is not equal to 0, x = -1 is not a root.

Trying x = 3:

(3^4 - 16(3)^3 + 108(3)^2 - 276(3) + 285) = 81 - 432 + 972 - 828 + 285 = 78

Since 78 is not equal to 0, x = 3 is not a root.

Trying x = -3:

((-3)^4 - 16(-3)^3 + 108(-3)^2 - 276(-3) + 285) = 81 + 432 + 972 + 828 + 285 = 2598

Since 2598 is not equal to 0, x = -3 is not a root.

Trying x = 5:

(5^4 - 16(5)^3 + 108(5)^2 - 276(5) + 285) = 625 - 2000 + 2700 - 1380 + 285 = 230

Since 230 is not equal to 0, x = 5 is not a root.

Trying x = -5:

((-5)^4 - 16(-5)^3 + 108(-5)^2 - 276(-5) + 285) = 625 + 2000 + 2700 + 1380 + 285 = 6990

Since 6990 is not equal to 0, x = -5 is not a root.

Trying x = 15:

(15^4 - 16(15)^3 + 108(15)^2 - 276(15) + 285) = 50625 - 43200 + 24300 - 4140 + 285 = 27570

Since 27570 is not equal to 0, x = 15 is not a root.

Trying x = -15:

((-15)^4 - 16(-15)^3 + 108(-15)^2 - 276(-15) + 285) = 50625 + 43200 + 24300 + 4140 + 285 = 122550

Since 122550 is not equal to 0, x = -15 is not a root.

Trying x = 19:

(19^4 - 16(19)^3 + 108(19)^2 - 276(19) + 285) = 130321 - 115456 + 38844 - 5244 + 285 = 105750

Since 105750 is not equal to 0, x = 19 is not a root.

Trying x = -19:

((-19)^4 - 16(-19)^3 + 108(-19)^2 - 276(-19) + 285) = 130321 + 115456 + 38844 + 5244 + 285 = 295550

Since 295550 is not equal to 0, x = -19 is not a root.

Trying x = 57:

(57^4 - 16(57)^3 + 108(57)^2 - 276(57) + 285) = 13997529 - 13069152 + 1966896 - 15732 + 285 = 100

Since 100 is not equal to 0, x = 57 is not a root.

Trying x = -57:

((-57)^4 - 16(-57)^3 + 108(-57)^2 - 276(-57) + 285) = 13997529 + 13069152 + 1966896 + 15732 + 285 = 29145994

Since 29145994 is not equal to 0, x = -57 is not a root.

Trying x = 95:

(95^4 - 16(95)^3 + 108(95)^2 - 276(95) + 285) = 81450625 - 121600000 + 9185400 - 26620 + 285 = 7125690

Since 7125690 is not equal to 0, x = 95 is not a root.

Trying x = -95:

((-95)^4 - 16(-95)^3 + 108(-95)^2 - 276(-95) + 285) = 81450625 + 121600000 + 9185400 + 26620 + 285 = 204506930

Since 204506930 is not equal to 0, x = -95 is not a root.

Trying x = 285:

(285^4 - 16(285)^3 + 108(285)^2 - 276(285) + 285) = 536723025 - 387420000 + 82207800 - 75900 + 285 = 168608210

Since 168608210 is not equal to 0, x = 285 is not a root.

Trying x = -285:

((-285)^4 - 16(-285)^3 + 108(-285)^2 - 276(-285) + 285) = 536723025 + 387420000 + 82207800 + 75900 + 285 = 1000000510

Since 1000000510 is not equal to 0, x = -285 is not a root.

Since none of the possible rational roots satisfy the equation, we can conclude that there are no rational solutions to the equation (x^2 - 6x + 18)(x^2 - 10x + 18) + 12x - 39 = 0.