Solve 1/4(2n+1)(n+4)-1/4n(2n+1)=2n+1

$$\frac { 1 } { 4 } ( 2 n + 1 ) ( n + 4 ) - \frac { 1 } { 4 } n ( 2 n + 1 ) = 2 n + 1$$

Solve for n (complex solution)

$$n\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4} by 2n+1.

$$\left(\frac{1}{2}n+\frac{1}{4}\right)\left(n+4\right)-\frac{1}{4}n\left(2n+1\right)=2n+1$$

Use the distributive property to multiply \frac{1}{2}n+\frac{1}{4} by n+4 and combine like terms.

$$\frac{1}{2}n^{2}+\frac{9}{4}n+1-\frac{1}{4}n\left(2n+1\right)=2n+1$$

Subtract 2n from both sides.

$$\frac{1}{2}n^{2}+\frac{9}{4}n+1-\frac{1}{4}n\left(2n+1\right)-2n=1$$

Use the distributive property to multiply -\frac{1}{4}n by 2n+1.

$$\frac{1}{2}n^{2}+\frac{9}{4}n+1-\frac{1}{2}n^{2}-\frac{1}{4}n-2n=1$$

Combine \frac{1}{2}n^{2} and -\frac{1}{2}n^{2} to get 0.

$$\frac{9}{4}n+1-\frac{1}{4}n-2n=1$$

Combine \frac{9}{4}n and -\frac{1}{4}n to get 2n.

$$2n+1-2n=1$$

Combine 2n and -2n to get 0.

$$1=1$$

Compare 1 and 1.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{C}$$

Solve for n

$$n\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4} by 2n+1.

$$\left(\frac{1}{2}n+\frac{1}{4}\right)\left(n+4\right)-\frac{1}{4}n\left(2n+1\right)=2n+1$$

Use the distributive property to multiply \frac{1}{2}n+\frac{1}{4} by n+4 and combine like terms.

$$\frac{1}{2}n^{2}+\frac{9}{4}n+1-\frac{1}{4}n\left(2n+1\right)=2n+1$$

Subtract 2n from both sides.

$$\frac{1}{2}n^{2}+\frac{9}{4}n+1-\frac{1}{4}n\left(2n+1\right)-2n=1$$

Use the distributive property to multiply -\frac{1}{4}n by 2n+1.

$$\frac{1}{2}n^{2}+\frac{9}{4}n+1-\frac{1}{2}n^{2}-\frac{1}{4}n-2n=1$$

Combine \frac{1}{2}n^{2} and -\frac{1}{2}n^{2} to get 0.

$$\frac{9}{4}n+1-\frac{1}{4}n-2n=1$$

Combine \frac{9}{4}n and -\frac{1}{4}n to get 2n.

$$2n+1-2n=1$$

Combine 2n and -2n to get 0.

$$1=1$$

Compare 1 and 1.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{R}$$