To solve the given equation, we can expand the expression (x+2)^4 using the binomial theorem, and then simplify the equation to solve for x.
Expanding (x+2)^4 using the binomial theorem:
(x+2)^4 = (4 choose 0)x^4(2)^0 + (4 choose 1)x^3(2)^1 + (4 choose 2)x^2(2)^2 + (4 choose 3)x^1(2)^3 + (4 choose 4)x^0*(2)^4
(x+2)^4 = 1x^41 + 4x^32 + 6x^24 + 4x8 + 1*16
(x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16
Now substitute back into the original equation:
(x+2)^4 + 2x^2 + 8x - 16 = 0(x^4 + 8x^3 + 24x^2 + 32x + 16) + 2x^2 + 8x - 16 = 0x^4 + 8x^3 + 26x^2 + 40x = 0
Now, we can factor out an x from the equation:
x(x^3 + 8x^2 + 26x + 40) = 0
At this point, we can see that x=0 is one solution. To find the other solutions, we can use numerical methods or try to factor the polynomial x^3 + 8x^2 + 26x + 40.
To solve the given equation, we can expand the expression (x+2)^4 using the binomial theorem, and then simplify the equation to solve for x.
Expanding (x+2)^4 using the binomial theorem:
(x+2)^4 = (4 choose 0)x^4(2)^0 + (4 choose 1)x^3(2)^1 + (4 choose 2)x^2(2)^2 + (4 choose 3)x^1(2)^3 + (4 choose 4)x^0*(2)^4
(x+2)^4 = 1x^41 + 4x^32 + 6x^24 + 4x8 + 1*16
(x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16
Now substitute back into the original equation:
(x+2)^4 + 2x^2 + 8x - 16 = 0
(x^4 + 8x^3 + 24x^2 + 32x + 16) + 2x^2 + 8x - 16 = 0
x^4 + 8x^3 + 26x^2 + 40x = 0
Now, we can factor out an x from the equation:
x(x^3 + 8x^2 + 26x + 40) = 0
At this point, we can see that x=0 is one solution. To find the other solutions, we can use numerical methods or try to factor the polynomial x^3 + 8x^2 + 26x + 40.