Solve -h(n+1)-2(n+1)^2-2(n+2)^2=-1

$$- h ( n + 1 ) - 2 ( n + 1 ) ^ { 2 } - 2 ( n + 2 ) ^ { 2 } = - 1$$

Solve for h

$$h=-\frac{\left(2n+3\right)^{2}}{n+1}$$

$$n\neq -1$$

Steps for Solving Linear Equation

Use the distributive property to multiply -h by n+1.

$$\left(-h\right)n-h-2\left(n+1\right)^{2}-2\left(n+2\right)^{2}=-1$$

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

$$\left(-h\right)n-h-2\left(n^{2}+2n+1\right)-2\left(n+2\right)^{2}=-1$$

Use the distributive property to multiply -2 by n^{2}+2n+1.

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

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

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

Use the distributive property to multiply -2 by n^{2}+4n+4.

$$\left(-h\right)n-h-2n^{2}-4n-2-2n^{2}-8n-8=-1$$

Combine -2n^{2} and -2n^{2} to get -4n^{2}.

$$\left(-h\right)n-h-4n^{2}-4n-2-8n-8=-1$$

Combine -4n and -8n to get -12n.

$$\left(-h\right)n-h-4n^{2}-12n-2-8=-1$$

Subtract 8 from -2 to get -10.

$$\left(-h\right)n-h-4n^{2}-12n-10=-1$$

Add 4n^{2} to both sides.

$$\left(-h\right)n-h-12n-10=-1+4n^{2}$$

Add 12n to both sides.

$$\left(-h\right)n-h-10=-1+4n^{2}+12n$$

Add 10 to both sides.

$$\left(-h\right)n-h=-1+4n^{2}+12n+10$$

Add -1 and 10 to get 9.

$$\left(-h\right)n-h=9+4n^{2}+12n$$

Reorder the terms.

$$-hn-h=4n^{2}+12n+9$$

Combine all terms containing h.

$$\left(-n-1\right)h=4n^{2}+12n+9$$

Divide both sides by -n-1.

$$\frac{\left(-n-1\right)h}{-n-1}=\frac{\left(2n+3\right)^{2}}{-n-1}$$

Dividing by -n-1 undoes the multiplication by -n-1.

$$h=\frac{\left(2n+3\right)^{2}}{-n-1}$$

Divide \left(2n+3\right)^{2} by -n-1.

$$h=-\frac{\left(2n+3\right)^{2}}{n+1}$$

Solve for n

$$n=\frac{-\sqrt{h\left(h+8\right)}-h-12}{8}$$

$$n=\frac{\sqrt{h\left(h+8\right)}-h-12}{8}\text{, }h\leq -8\text{ or }h\geq 0$$