Solve 2^2x+1-(2m-3)*2^x+1+(7-2m)=0

$$2 ^ { 2 x + 1 } - ( 2 m - 3 ) \cdot 2 ^ { x + 1 } + ( 7 - 2 m ) = 0$$

Solve for m

$$m=\frac{2\times 4^{x}+6\times 2^{x}+7}{2\left(2^{x+1}+1\right)}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m-3 by 2^{x+1}.

$$2^{2x+1}-\left(2m\times 2^{x+1}-3\times 2^{x+1}\right)+7-2m=0$$

To find the opposite of 2m\times 2^{x+1}-3\times 2^{x+1}, find the opposite of each term.

$$2^{2x+1}-2m\times 2^{x+1}+3\times 2^{x+1}+7-2m=0$$

Subtract 2^{2x+1} from both sides. Anything subtracted from zero gives its negation.

$$-2m\times 2^{x+1}+3\times 2^{x+1}+7-2m=-2^{2x+1}$$

Subtract 3\times 2^{x+1} from both sides.

$$-2m\times 2^{x+1}+7-2m=-2^{2x+1}-3\times 2^{x+1}$$

Subtract 7 from both sides.

$$-2m\times 2^{x+1}-2m=-2^{2x+1}-3\times 2^{x+1}-7$$

Combine all terms containing m.

$$\left(-2\times 2^{x+1}-2\right)m=-2^{2x+1}-3\times 2^{x+1}-7$$

The equation is in standard form.

$$\left(-2\times 2^{x+1}-2\right)m=-3\times 2^{x+1}-2^{2x+1}-7$$

Divide both sides by -2\times 2^{x+1}-2.

$$\frac{\left(-2\times 2^{x+1}-2\right)m}{-2\times 2^{x+1}-2}=\frac{-2\times 4^{x}-6\times 2^{x}-7}{-2\times 2^{x+1}-2}$$

Dividing by -2\times 2^{x+1}-2 undoes the multiplication by -2\times 2^{x+1}-2.

$$m=\frac{-2\times 4^{x}-6\times 2^{x}-7}{-2\times 2^{x+1}-2}$$

Divide -2\times 4^{x}-6\times 2^{x}-7 by -2\times 2^{x+1}-2.

$$m=\frac{2\times 4^{x}+6\times 2^{x}+7}{2\left(2\times 2^{x}+1\right)}$$

Solve for x

$$\left\{\begin{matrix}x=\log_{2}\left(\frac{\sqrt{4m^{2}-8m-5}}{2}+m-\frac{3}{2}\right)\text{, }&m\geq \frac{5}{2}\\x=\log_{2}\left(-\frac{\sqrt{4m^{2}-8m-5}}{2}+m-\frac{3}{2}\right)\text{, }&m\geq \frac{5}{2}\text{ and }m<\frac{7}{2}\end{matrix}\right.$$