Solve (3x+5)^2-(x+2)^2=(2x+3)^2+(2x+1)^2+411x

$$( 3 x + 5 ) ^ { 2 } - ( x + 2 ) ^ { 2 } = ( 2 x + 3 ) ^ { 2 } + ( 2 x + 1 ) ^ { 2 } + 411 x$$

Solve for x

$$x=\frac{11}{401}\approx 0.027431421$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+5\right)^{2}.

$$9x^{2}+30x+25-\left(x+2\right)^{2}=\left(2x+3\right)^{2}+\left(2x+1\right)^{2}+411x$$

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

$$9x^{2}+30x+25-\left(x^{2}+4x+4\right)=\left(2x+3\right)^{2}+\left(2x+1\right)^{2}+411x$$

To find the opposite of x^{2}+4x+4, find the opposite of each term.

$$9x^{2}+30x+25-x^{2}-4x-4=\left(2x+3\right)^{2}+\left(2x+1\right)^{2}+411x$$

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

$$8x^{2}+30x+25-4x-4=\left(2x+3\right)^{2}+\left(2x+1\right)^{2}+411x$$

Combine 30x and -4x to get 26x.

$$8x^{2}+26x+25-4=\left(2x+3\right)^{2}+\left(2x+1\right)^{2}+411x$$

Subtract 4 from 25 to get 21.

$$8x^{2}+26x+21=\left(2x+3\right)^{2}+\left(2x+1\right)^{2}+411x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$8x^{2}+26x+21=4x^{2}+12x+9+\left(2x+1\right)^{2}+411x$$

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

$$8x^{2}+26x+21=4x^{2}+12x+9+4x^{2}+4x+1+411x$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}+26x+21=8x^{2}+12x+9+4x+1+411x$$

Combine 12x and 4x to get 16x.

$$8x^{2}+26x+21=8x^{2}+16x+9+1+411x$$

Add 9 and 1 to get 10.

$$8x^{2}+26x+21=8x^{2}+16x+10+411x$$

Combine 16x and 411x to get 427x.

$$8x^{2}+26x+21=8x^{2}+427x+10$$

Subtract 8x^{2} from both sides.

$$8x^{2}+26x+21-8x^{2}=427x+10$$

Combine 8x^{2} and -8x^{2} to get 0.

$$26x+21=427x+10$$

Subtract 427x from both sides.

$$26x+21-427x=10$$

Combine 26x and -427x to get -401x.

$$-401x+21=10$$

Subtract 21 from both sides.

$$-401x=10-21$$

Subtract 21 from 10 to get -11.

$$-401x=-11$$

Divide both sides by -401.

$$x=\frac{-11}{-401}$$

Fraction \frac{-11}{-401} can be simplified to \frac{11}{401} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{11}{401}$$