Solve -(2+N)^2+4(2+n)=-6

$$- ( 2 + N ) ^ { 2 } + 4 ( 2 + n ) = - 6$$

Solve for n

$$n=\frac{\left(N+2\right)^{2}-14}{4}$$

Steps for Solving Linear Equation

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

$$-\left(4+4N+N^{2}\right)+4\left(2+n\right)=-6$$

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

$$-4-4N-N^{2}+4\left(2+n\right)=-6$$

Use the distributive property to multiply 4 by 2+n.

$$-4-4N-N^{2}+8+4n=-6$$

Add -4 and 8 to get 4.

$$4-4N-N^{2}+4n=-6$$

Subtract 4 from both sides.

$$-4N-N^{2}+4n=-6-4$$

Subtract 4 from -6 to get -10.

$$-4N-N^{2}+4n=-10$$

Add 4N to both sides.

$$-N^{2}+4n=-10+4N$$

Add N^{2} to both sides.

$$4n=-10+4N+N^{2}$$

The equation is in standard form.

$$4n=N^{2}+4N-10$$

Divide both sides by 4.

$$\frac{4n}{4}=\frac{N^{2}+4N-10}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$n=\frac{N^{2}+4N-10}{4}$$

Divide -10+4N+N^{2} by 4.

$$n=\frac{N^{2}}{4}+N-\frac{5}{2}$$

Solve for N (complex solution)

$$N=-\sqrt{4n+14}-2$$

$$N=\sqrt{4n+14}-2$$

Solve for N

$$N=-\sqrt{4n+14}-2$$

$$N=\sqrt{4n+14}-2\text{, }n\geq -\frac{7}{2}$$