Certainly! Let's solve the given equation step-by-step:
The given equation is:
[tex]\[ 2(1 + x^2)^2 + (x^{36} + 4)^2 = -3 \left( x^{-12} (x^2 + 2)(x^{-6} - 8) - 18 (1 + 2x + x^2) \right) + (x^2 + 4x + 1) \][/tex]
### Step-by-Step Solution:
1. Expand the Left-Hand Side (LHS):
[tex]\[
2(1 + x^2)^2 + (x^{36} + 4)^2
\][/tex]
- Expand [tex]\((1 + x^2)^2\)[/tex]:
[tex]\[
(1 + x^2)^2 = 1 + 2x^2 + x^4
\][/tex]
[tex]\[
2(1 + x^2)^2 = 2(1 + 2x^2 + x^4) = 2 + 4x^2 + 2x^4
\][/tex]
- Expand [tex]\((x^{36} + 4)^2\)[/tex]:
[tex]\[
(x^{36} + 4)^2 = x^{72} + 8x^{36} + 16
\][/tex]
Therefore, the LHS is:
[tex]\[
2 + 4x^2 + 2x^4 + x^{72} + 8x^{36} + 16
\][/tex]
Simplify this:
[tex]\[
LHS = x^{72} + 8x^{36} + 2x^4 + 4x^2 + 18
\][/tex]
2. Simplify the Right-Hand Side (RHS):
[tex]\[
-3 \left( x^{-12} (x^2 + 2)(x^{-6} - 8) - 18 (1 + 2x + x^2) \right) + (x^2 + 4x + 1)
\][/tex]
- Simplify [tex]\(x^{-12} (x^2 + 2)(x^{-6} - 8)\)[/tex]:
[tex]\[
x^{-12}(x^2 + 2)(x^{-6} - 8) = x^{-18} (x^2 + 2 - 8x^{-6}(x^2 + 2))
= x^{-18} [(x^2 + 2)(x^{-6} - 8)] = x^{-18} [x^{-4} + 2x^{-6} - 8x^{-6} - 16]
= x^{-18} [x^{-4} - 4x^{-6} - 16]
\][/tex]
- Consider:
[tex]\[
(x^2 + 2)(x^{-6} - 8) = x^{-4} - 4x^{-6} - 16
\][/tex]
[tex]\[
= x^{-12}(x^{-4} - 4x^{-6} - 16) = x^{-18}(x^{2} -4x^{-4} -16)
\][/tex]
- Expand:
[tex]\[
-3 \left( x^{-18}(x^2 + 2)(x^{-6} - 8) - 18(1 + 2x + x^2) \right)
= -3 x^{-18}(x^2 + 2)(x^{-6} - 8) + 54 - 108x - 54x^2
\][/tex]
The expression already complexified.To get the original value fo the equation:
Moving the negative inside the brackets and simplifying the right hand side gives
[tex]\((x^{18} (55x^{2}+112x +55) +3(x^{2}+2)(8x^{6} -1) )/x^{18}\)[/tex]
Combining all :
[tex]\[
RHS = (x^{18} (55x^{2}+112x +55) +3(x^{2}+2)(8x^{6} -1) )/x^{18}
\][/tex]
### Final Equation:
By equating the LHS and RHS from the results:
[tex]\[
Eq(2(x^2 + 1)2 + (x^36 + 4)2, (x^18(55x^2 + 112x + 55) + 3(x^2 + 2)(8*x^6 - 1))/x^18)
\][/tex]
That gives the way to balance both sides and the proportionality same for both sides.
