Solve 50+left(2x^2-3x+1right)^2-left((x+2)(x-2)+(x+1)(x-4)right)^2=(4x-7)(9x+2)

$$50+ { \left(2 { x }^{ 2 } -3x+1 \right) }^{ 2 } - { \left((x+2)(x-2)+(x+1)(x-4) \right) }^{ 2 } =(4x-7)(9x+2)$$

Solve for x

$$x=-1$$

Steps for Solving Linear Equation

Square 2x^{2}-3x+1.

$$50+4x^{4}-12x^{3}+13x^{2}-6x+1-\left(\left(x+2\right)\left(x-2\right)+\left(x+1\right)\left(x-4\right)\right)^{2}=\left(4x-7\right)\left(9x+2\right)$$

Add 50 and 1 to get 51.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-\left(\left(x+2\right)\left(x-2\right)+\left(x+1\right)\left(x-4\right)\right)^{2}=\left(4x-7\right)\left(9x+2\right)$$

Consider \left(x+2\right)\left(x-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-\left(x^{2}-4+\left(x+1\right)\left(x-4\right)\right)^{2}=\left(4x-7\right)\left(9x+2\right)$$

Use the distributive property to multiply x+1 by x-4 and combine like terms.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-\left(x^{2}-4+x^{2}-3x-4\right)^{2}=\left(4x-7\right)\left(9x+2\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-\left(2x^{2}-4-3x-4\right)^{2}=\left(4x-7\right)\left(9x+2\right)$$

Subtract 4 from -4 to get -8.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-\left(2x^{2}-8-3x\right)^{2}=\left(4x-7\right)\left(9x+2\right)$$

Square 2x^{2}-8-3x.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-\left(4x^{4}-12x^{3}-23x^{2}+48x+64\right)=\left(4x-7\right)\left(9x+2\right)$$

To find the opposite of 4x^{4}-12x^{3}-23x^{2}+48x+64, find the opposite of each term.

$$51+4x^{4}-12x^{3}+13x^{2}-6x-4x^{4}+12x^{3}+23x^{2}-48x-64=\left(4x-7\right)\left(9x+2\right)$$

Combine 4x^{4} and -4x^{4} to get 0.

$$51-12x^{3}+13x^{2}-6x+12x^{3}+23x^{2}-48x-64=\left(4x-7\right)\left(9x+2\right)$$

Combine -12x^{3} and 12x^{3} to get 0.

$$51+13x^{2}-6x+23x^{2}-48x-64=\left(4x-7\right)\left(9x+2\right)$$

Combine 13x^{2} and 23x^{2} to get 36x^{2}.

$$51+36x^{2}-6x-48x-64=\left(4x-7\right)\left(9x+2\right)$$

Combine -6x and -48x to get -54x.

$$51+36x^{2}-54x-64=\left(4x-7\right)\left(9x+2\right)$$

Subtract 64 from 51 to get -13.

$$-13+36x^{2}-54x=\left(4x-7\right)\left(9x+2\right)$$

Use the distributive property to multiply 4x-7 by 9x+2 and combine like terms.

$$-13+36x^{2}-54x=36x^{2}-55x-14$$

Subtract 36x^{2} from both sides.

$$-13+36x^{2}-54x-36x^{2}=-55x-14$$

Combine 36x^{2} and -36x^{2} to get 0.

$$-13-54x=-55x-14$$

Add 55x to both sides.

$$-13-54x+55x=-14$$

Combine -54x and 55x to get x.

$$-13+x=-14$$

Add 13 to both sides.

$$x=-14+13$$

Add -14 and 13 to get -1.

$$x=-1$$