Solve (2x+1)^2+x(2x+1)+AH=5x(2x+1)

$$( 2 x + 1 ) ^ { 2 } + x ( 2 x + 1 ) + A H = 5 x ( 2 x + 1 )$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=\frac{4x^{2}-1}{H}\text{, }&H\neq 0\\A\in \mathrm{C}\text{, }&\left(x=-\frac{1}{2}\text{ or }x=\frac{1}{2}\right)\text{ and }H=0\end{matrix}\right.$$

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}.

$$4x^{2}+4x+1+x\left(2x+1\right)+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply x by 2x+1.

$$4x^{2}+4x+1+2x^{2}+x+AH=5x\left(2x+1\right)$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$6x^{2}+4x+1+x+AH=5x\left(2x+1\right)$$

Combine 4x and x to get 5x.

$$6x^{2}+5x+1+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply 5x by 2x+1.

$$6x^{2}+5x+1+AH=10x^{2}+5x$$

Subtract 6x^{2} from both sides.

$$5x+1+AH=10x^{2}+5x-6x^{2}$$

Combine 10x^{2} and -6x^{2} to get 4x^{2}.

$$5x+1+AH=4x^{2}+5x$$

Subtract 5x from both sides.

$$1+AH=4x^{2}+5x-5x$$

Combine 5x and -5x to get 0.

$$1+AH=4x^{2}$$

Subtract 1 from both sides.

$$AH=4x^{2}-1$$

The equation is in standard form.

$$HA=4x^{2}-1$$

Divide both sides by H.

$$\frac{HA}{H}=\frac{4x^{2}-1}{H}$$

Dividing by H undoes the multiplication by H.

$$A=\frac{4x^{2}-1}{H}$$

Solve for H (complex solution)

$$\left\{\begin{matrix}H=\frac{4x^{2}-1}{A}\text{, }&A\neq 0\\H\in \mathrm{C}\text{, }&\left(x=-\frac{1}{2}\text{ or }x=\frac{1}{2}\right)\text{ and }A=0\end{matrix}\right.$$

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}.

$$4x^{2}+4x+1+x\left(2x+1\right)+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply x by 2x+1.

$$4x^{2}+4x+1+2x^{2}+x+AH=5x\left(2x+1\right)$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$6x^{2}+4x+1+x+AH=5x\left(2x+1\right)$$

Combine 4x and x to get 5x.

$$6x^{2}+5x+1+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply 5x by 2x+1.

$$6x^{2}+5x+1+AH=10x^{2}+5x$$

Subtract 6x^{2} from both sides.

$$5x+1+AH=10x^{2}+5x-6x^{2}$$

Combine 10x^{2} and -6x^{2} to get 4x^{2}.

$$5x+1+AH=4x^{2}+5x$$

Subtract 5x from both sides.

$$1+AH=4x^{2}+5x-5x$$

Combine 5x and -5x to get 0.

$$1+AH=4x^{2}$$

Subtract 1 from both sides.

$$AH=4x^{2}-1$$

Divide both sides by A.

$$\frac{AH}{A}=\frac{4x^{2}-1}{A}$$

Dividing by A undoes the multiplication by A.

$$H=\frac{4x^{2}-1}{A}$$

Solve for A

$$\left\{\begin{matrix}A=\frac{4x^{2}-1}{H}\text{, }&H\neq 0\\A\in \mathrm{R}\text{, }&H=0\text{ and }|x|=\frac{1}{2}\end{matrix}\right.$$

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}.

$$4x^{2}+4x+1+x\left(2x+1\right)+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply x by 2x+1.

$$4x^{2}+4x+1+2x^{2}+x+AH=5x\left(2x+1\right)$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$6x^{2}+4x+1+x+AH=5x\left(2x+1\right)$$

Combine 4x and x to get 5x.

$$6x^{2}+5x+1+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply 5x by 2x+1.

$$6x^{2}+5x+1+AH=10x^{2}+5x$$

Subtract 6x^{2} from both sides.

$$5x+1+AH=10x^{2}+5x-6x^{2}$$

Combine 10x^{2} and -6x^{2} to get 4x^{2}.

$$5x+1+AH=4x^{2}+5x$$

Subtract 5x from both sides.

$$1+AH=4x^{2}+5x-5x$$

Combine 5x and -5x to get 0.

$$1+AH=4x^{2}$$

Subtract 1 from both sides.

$$AH=4x^{2}-1$$

The equation is in standard form.

$$HA=4x^{2}-1$$

Divide both sides by H.

$$\frac{HA}{H}=\frac{4x^{2}-1}{H}$$

Dividing by H undoes the multiplication by H.

$$A=\frac{4x^{2}-1}{H}$$

Solve for H

$$\left\{\begin{matrix}H=\frac{4x^{2}-1}{A}\text{, }&A\neq 0\\H\in \mathrm{R}\text{, }&A=0\text{ and }|x|=\frac{1}{2}\end{matrix}\right.$$

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}.

$$4x^{2}+4x+1+x\left(2x+1\right)+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply x by 2x+1.

$$4x^{2}+4x+1+2x^{2}+x+AH=5x\left(2x+1\right)$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$6x^{2}+4x+1+x+AH=5x\left(2x+1\right)$$

Combine 4x and x to get 5x.

$$6x^{2}+5x+1+AH=5x\left(2x+1\right)$$

Use the distributive property to multiply 5x by 2x+1.

$$6x^{2}+5x+1+AH=10x^{2}+5x$$

Subtract 6x^{2} from both sides.

$$5x+1+AH=10x^{2}+5x-6x^{2}$$

Combine 10x^{2} and -6x^{2} to get 4x^{2}.

$$5x+1+AH=4x^{2}+5x$$

Subtract 5x from both sides.

$$1+AH=4x^{2}+5x-5x$$

Combine 5x and -5x to get 0.

$$1+AH=4x^{2}$$

Subtract 1 from both sides.

$$AH=4x^{2}-1$$

Divide both sides by A.

$$\frac{AH}{A}=\frac{4x^{2}-1}{A}$$

Dividing by A undoes the multiplication by A.

$$H=\frac{4x^{2}-1}{A}$$