Solve 2(2x+1)^2-(3x+1)(3x-1)=35-x^2

$$2 ( 2 x + 1 ) ^ { 2 } - ( 3 x + 1 ) ( 3 x - 1 ) = 35 - x ^ { 2 }$$

Solve for x

$$x=4$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$2\left(4x^{2}+4x+1\right)-\left(3x+1\right)\left(3x-1\right)=35-x^{2}$$

Use the distributive property to multiply 2 by 4x^{2}+4x+1.

$$8x^{2}+8x+2-\left(3x+1\right)\left(3x-1\right)=35-x^{2}$$

Consider \left(3x+1\right)\left(3x-1\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 1.

$$8x^{2}+8x+2-\left(\left(3x\right)^{2}-1\right)=35-x^{2}$$

Expand \left(3x\right)^{2}.

$$8x^{2}+8x+2-\left(3^{2}x^{2}-1\right)=35-x^{2}$$

Calculate 3 to the power of 2 and get 9.

$$8x^{2}+8x+2-\left(9x^{2}-1\right)=35-x^{2}$$

To find the opposite of 9x^{2}-1, find the opposite of each term.

$$8x^{2}+8x+2-9x^{2}+1=35-x^{2}$$

Combine 8x^{2} and -9x^{2} to get -x^{2}.

$$-x^{2}+8x+2+1=35-x^{2}$$

Add 2 and 1 to get 3.

$$-x^{2}+8x+3=35-x^{2}$$

Add x^{2} to both sides.

$$-x^{2}+8x+3+x^{2}=35$$

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

$$8x+3=35$$

Subtract 3 from both sides.

$$8x=35-3$$

Subtract 3 from 35 to get 32.

$$8x=32$$

Divide both sides by 8.

$$x=\frac{32}{8}$$

Divide 32 by 8 to get 4.

$$x=4$$