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① The substitution x+10 =t is brilliant . In 8:24 Math Booster found : t⁴+6t²-352=0. Let t²=w and then you have to solve a very easy equation. w²+6w-352=0 Personally I use the discriminant formula. D=36+4•352=1444=38²>0 So w=(-6±38)/2 => w=16 or w=-22 => t²=16 or t²=-22 ……….. ② Another approach (more difficult): (x+9)⁴+(x+11)⁴=706 (x+9)⁴+(x+9+2)=706 (let x+9=t) => t⁴+(t+2)⁴=706 …….. t⁴+4t³+12t²+16t-345=0 (1) The possible integral roots of the equation are the divisors of -345. Hence : ±1,±3,±5,±23 *Using Horner’s method* 1 4 12 16 -345 ρ=3 # 3 21 99 345 1 7 33 115 0 So (1) => (t-3)(t³+7t²+33t+115)=0 (2) 1 7 33 115 ρ=-5 # -5 -10 -115 1 2 23 0 (2)=> (t-3)(t+5)(t²+2t+23)=0 ….. etc
