(x^2+3x-4)^3+(2x^2-5x+3)^3=(3x^2-2x-1)^3

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вопрос

Expanding both sides:

Left side:
(x^2+3x-4)^3 = (x^2+3x-4)(x^2+3x-4)(x^2+3x-4)
Expanding each set of parentheses:
= (x^2+3x-4)(x^2+3x-4) = [(x^4+6x^3+9x^2-4x^2-24x+16)(x^2+3x-4)]
= (x^6+6x^5+9x^4-4x^4-24x^3+16x^2+3x^5+18x^4+27x^3-12x^3-72x^2+48x-4x^2-24x+16)
= x^6+9x^5+23x^4+x^3-32x^2+24x+16

(2x^2-5x+3)^3 = (2x^2-5x+3)(2x^2-5x+3)(2x^2-5x+3)
Expanding each set of parentheses:
= [(4x^4-10x^3+6x^2-10x^3+25x^2-15x+6x^2-15x+9)(2x^2-5x+3)]
= [(4x^4-20x^3+31x^2-30x+9)(2x^2-5x+3)]
= 8x^6-40x^5+62x^4-60x^2+18x^4-90x^3+93x^2-45x+27x^2-135x+81
= 8x^6-40x^5+80x^4-90x^3+60x^2-180x+81

Now adding these two results together:
x^6+9x^5+23x^4+x^3-32x^2+24x+16 + 8x^6-40x^5+80x^4-90x^3+60x^2-180x+81
= 9x^6-31x^5+103x^4-89x^3+28x^2-156x+97

And simplifying the right side:
(3x^2-2x-1)^3 = (3x^2-2x-1)(3x^2-2x-1)(3x^2-2x-1)
= [(9x^4-6x^3-3x^3+2x^2-2x^2+1)(3x^2-2x-1)]
= [(9x^4-9x^3+2x^2+1)(3x^2-2x-1)]
= 27x^6-27x^5+6x^4+3x^2-18x^3+18x^2-4x-2x-1
= 27x^6-27x^5+6x^4-18x^3+21x^2-6x-1

Therefore, the two sides are equal.