(1+4x)(a*x*x+bx+c)+(x-x*x)(2ax+b)-(x^2+x^3)(2a)=0

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Solution for (1+4x)(a*x*x+bx+c)+(x-x*x)(2ax+b)-(x^2+x^3)(2a)=0 equation: 


Simplifying
(1 + 4x)(a * x * x + bx + c) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0

Multiply a * x
(1 + 4x)(ax * x + bx + c) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0

Multiply ax * x
(1 + 4x)(ax2 + bx + c) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0

Multiply (1 + 4x) * (ax2 + bx + c)
(1(ax2 + bx + c) + 4x * (ax2 + bx + c)) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0
((ax2 * 1 + bx * 1 + c * 1) + 4x * (ax2 + bx + c)) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0
((1ax2 + 1bx + 1c) + 4x * (ax2 + bx + c)) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0
(1ax2 + 1bx + 1c + (ax2 * 4x + bx * 4x + c * 4x)) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0
(1ax2 + 1bx + 1c + (4ax3 + 4bx2 + 4cx)) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0

Reorder the terms:
(1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0
(1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx) + (x + -1x * x)(2ax + b) + -1(x2 + x3)(2a) = 0

Multiply x * x
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + (x + -1x2)(2ax + b) + -1(x2 + x3)(2a) = 0

Multiply (x + -1x2) * (2ax + b)
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + (x(2ax + b) + -1x2 * (2ax + b)) + -1(x2 + x3)(2a) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + ((2ax * x + b * x) + -1x2 * (2ax + b)) + -1(x2 + x3)(2a) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + ((2ax2 + bx) + -1x2 * (2ax + b)) + -1(x2 + x3)(2a) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + (2ax2 + bx + (2ax * -1x2 + b * -1x2)) + -1(x2 + x3)(2a) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + (2ax2 + bx + (-2ax3 + -1bx2)) + -1(x2 + x3)(2a) = 0

Reorder the terms:
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + (2ax2 + -2ax3 + bx + -1bx2) + -1(x2 + x3)(2a) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + (2ax2 + -2ax3 + bx + -1bx2) + -1(x2 + x3)(2a) = 0

Remove parenthesis around (2a)
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + 2ax2 + -2ax3 + bx + -1bx2 + -1(x2 + x3) * 2a = 0

Reorder the terms for easier multiplication:
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + 2ax2 + -2ax3 + bx + -1bx2 + -1 * 2a(x2 + x3) = 0

Multiply -1 * 2
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + 2ax2 + -2ax3 + bx + -1bx2 + -2a(x2 + x3) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + 2ax2 + -2ax3 + bx + -1bx2 + (x2 * -2a + x3 * -2a) = 0
1ax2 + 4ax3 + 1bx + 4bx2 + 1c + 4cx + 2ax2 + -2ax3 + bx + -1bx2 + (-2ax2 + -2ax3) = 0

Reorder the terms:
1ax2 + 2ax2 + -2ax2 + 4ax3 + -2ax3 + -2ax3 + 1bx + bx + 4bx2 + -1bx2 + 1c + 4cx = 0

Combine like terms: 1ax2 + 2ax2 = 3ax2
3ax2 + -2ax2 + 4ax3 + -2ax3 + -2ax3 + 1bx + bx + 4bx2 + -1bx2 + 1c + 4cx = 0

Combine like terms: 3ax2 + -2ax2 = 1ax2
1ax2 + 4ax3 + -2ax3 + -2ax3 + 1bx + bx + 4bx2 + -1bx2 + 1c + 4cx = 0

Combine like terms: 4ax3 + -2ax3 = 2ax3
1ax2 + 2ax3 + -2ax3 + 1bx + bx + 4bx2 + -1bx2 + 1c + 4cx = 0

Combine like terms: 2ax3 + -2ax3 = 0
1ax2 + 0 + 1bx + bx + 4bx2 + -1bx2 + 1c + 4cx = 0
1ax2 + 1bx + bx + 4bx2 + -1bx2 + 1c + 4cx = 0

Combine like terms: 1bx + bx = 2bx
1ax2 + 2bx + 4bx2 + -1bx2 + 1c + 4cx = 0

Combine like terms: 4bx2 + -1bx2 = 3bx2
1ax2 + 2bx + 3bx2 + 1c + 4cx = 0

Solving
1ax2 + 2bx + 3bx2 + 1c + 4cx = 0

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-2bx' to each side of the equation.
1ax2 + 2bx + 3bx2 + 1c + -2bx + 4cx = 0 + -2bx

Reorder the terms:
1ax2 + 2bx + -2bx + 3bx2 + 1c + 4cx = 0 + -2bx

Combine like terms: 2bx + -2bx = 0
1ax2 + 0 + 3bx2 + 1c + 4cx = 0 + -2bx
1ax2 + 3bx2 + 1c + 4cx = 0 + -2bx
Remove the zero:
1ax2 + 3bx2 + 1c + 4cx = -2bx

Add '-3bx2' to each side of the equation.
1ax2 + 3bx2 + 1c + -3bx2 + 4cx = -2bx + -3bx2

Reorder the terms:
1ax2 + 3bx2 + -3bx2 + 1c + 4cx = -2bx + -3bx2

Combine like terms: 3bx2 + -3bx2 = 0
1ax2 + 0 + 1c + 4cx = -2bx + -3bx2
1ax2 + 1c + 4cx = -2bx + -3bx2

Add '-1c' to each side of the equation.
1ax2 + 1c + -1c + 4cx = -2bx + -3bx2 + -1c

Combine like terms: 1c + -1c = 0
1ax2 + 0 + 4cx = -2bx + -3bx2 + -1c
1ax2 + 4cx = -2bx + -3bx2 + -1c

Add '-4cx' to each side of the equation.
1ax2 + 4cx + -4cx = -2bx + -3bx2 + -1c + -4cx

Combine like terms: 4cx + -4cx = 0
1ax2 + 0 = -2bx + -3bx2 + -1c + -4cx
1ax2 = -2bx + -3bx2 + -1c + -4cx

Divide each side by '1x2'.
a = -2bx-1 + -3b + -1cx-2 + -4cx-1

Simplifying
a = -2bx-1 + -3b + -1cx-2 + -4cx-1

Reorder the terms:
a = -3b + -2bx-1 + -1cx-2 + -4cx-1

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