Solve 2(n^2Y-4n-4)+4(nY-4)+10Y=16e^-3n

$$2 ( n ^ { 2 } Y - 4 n - 4 ) + 4 ( n Y - 4 ) + 10 Y = 16 e ^ { - 3 n }$$

Solve for Y (complex solution)

$$Y=\frac{4\left(ne^{3n}+3e^{3n}+2\right)}{\left(n^{2}+2n+5\right)e^{3n}}$$

$$n\neq -1+2i\text{ and }n\neq -1-2i$$

Steps for Solving Linear Equation

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

$$2n^{2}Y-8n-8+4\left(nY-4\right)+10Y=16e^{-3n}$$

Use the distributive property to multiply 4 by nY-4.

$$2n^{2}Y-8n-8+4nY-16+10Y=16e^{-3n}$$

Subtract 16 from -8 to get -24.

$$2n^{2}Y-8n-24+4nY+10Y=16e^{-3n}$$

Add 8n to both sides.

$$2n^{2}Y-24+4nY+10Y=16e^{-3n}+8n$$

Add 24 to both sides.

$$2n^{2}Y+4nY+10Y=16e^{-3n}+8n+24$$

Combine all terms containing Y.

$$\left(2n^{2}+4n+10\right)Y=16e^{-3n}+8n+24$$

The equation is in standard form.

$$\left(2n^{2}+4n+10\right)Y=\frac{16}{e^{3n}}+8n+24$$

Divide both sides by 2n^{2}+4n+10.

$$\frac{\left(2n^{2}+4n+10\right)Y}{2n^{2}+4n+10}=\frac{\frac{16}{e^{3n}}+8n+24}{2n^{2}+4n+10}$$

Dividing by 2n^{2}+4n+10 undoes the multiplication by 2n^{2}+4n+10.

$$Y=\frac{\frac{16}{e^{3n}}+8n+24}{2n^{2}+4n+10}$$

Divide 24+8n+\frac{16}{e^{3n}} by 2n^{2}+4n+10.

$$Y=\frac{4\left(ne^{3n}+3e^{3n}+2\right)}{\left(n^{2}+2n+5\right)e^{3n}}$$

Solve for Y

$$Y=\frac{4\left(ne^{3n}+3e^{3n}+2\right)}{\left(n^{2}+2n+5\right)e^{3n}}$$

Steps for Solving Linear Equation

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

$$2n^{2}Y-8n-8+4\left(nY-4\right)+10Y=16e^{-3n}$$

Use the distributive property to multiply 4 by nY-4.

$$2n^{2}Y-8n-8+4nY-16+10Y=16e^{-3n}$$

Subtract 16 from -8 to get -24.

$$2n^{2}Y-8n-24+4nY+10Y=16e^{-3n}$$

Add 8n to both sides.

$$2n^{2}Y-24+4nY+10Y=16e^{-3n}+8n$$

Add 24 to both sides.

$$2n^{2}Y+4nY+10Y=16e^{-3n}+8n+24$$

Combine all terms containing Y.

$$\left(2n^{2}+4n+10\right)Y=16e^{-3n}+8n+24$$

The equation is in standard form.

$$\left(2n^{2}+4n+10\right)Y=\frac{16}{e^{3n}}+8n+24$$

Divide both sides by 2n^{2}+4n+10.

$$\frac{\left(2n^{2}+4n+10\right)Y}{2n^{2}+4n+10}=\frac{\frac{16}{e^{3n}}+8n+24}{2n^{2}+4n+10}$$

Dividing by 2n^{2}+4n+10 undoes the multiplication by 2n^{2}+4n+10.

$$Y=\frac{\frac{16}{e^{3n}}+8n+24}{2n^{2}+4n+10}$$

Divide 24+8n+\frac{16}{e^{3n}} by 2n^{2}+4n+10.

$$Y=\frac{4\left(ne^{3n}+3e^{3n}+2\right)}{\left(n^{2}+2n+5\right)e^{3n}}$$