Solve 8t(x+2)^2-3(x+2)=0*4

$$8 t ( x + 2 ) ^ { 2 } - 3 ( x + 2 ) = 0 \cdot 4$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=\frac{3}{8\left(x+2\right)}\text{, }&x\neq -2\\t\in \mathrm{C}\text{, }&x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$8t\left(x^{2}+4x+4\right)-3\left(x+2\right)=0\times 4$$

Use the distributive property to multiply 8t by x^{2}+4x+4.

$$8tx^{2}+32tx+32t-3\left(x+2\right)=0\times 4$$

Use the distributive property to multiply -3 by x+2.

$$8tx^{2}+32tx+32t-3x-6=0\times 4$$

Multiply 0 and 4 to get 0.

$$8tx^{2}+32tx+32t-3x-6=0$$

Add 3x to both sides. Anything plus zero gives itself.

$$8tx^{2}+32tx+32t-6=3x$$

Add 6 to both sides.

$$8tx^{2}+32tx+32t=3x+6$$

Combine all terms containing t.

$$\left(8x^{2}+32x+32\right)t=3x+6$$

Divide both sides by 8x^{2}+32x+32.

$$\frac{\left(8x^{2}+32x+32\right)t}{8x^{2}+32x+32}=\frac{3x+6}{8x^{2}+32x+32}$$

Dividing by 8x^{2}+32x+32 undoes the multiplication by 8x^{2}+32x+32.

$$t=\frac{3x+6}{8x^{2}+32x+32}$$

Divide 6+3x by 8x^{2}+32x+32.

$$t=\frac{3}{8\left(x+2\right)}$$

Solve for t

$$\left\{\begin{matrix}t=\frac{3}{8\left(x+2\right)}\text{, }&x\neq -2\\t\in \mathrm{R}\text{, }&x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$8t\left(x^{2}+4x+4\right)-3\left(x+2\right)=0\times 4$$

Use the distributive property to multiply 8t by x^{2}+4x+4.

$$8tx^{2}+32tx+32t-3\left(x+2\right)=0\times 4$$

Use the distributive property to multiply -3 by x+2.

$$8tx^{2}+32tx+32t-3x-6=0\times 4$$

Multiply 0 and 4 to get 0.

$$8tx^{2}+32tx+32t-3x-6=0$$

Add 3x to both sides. Anything plus zero gives itself.

$$8tx^{2}+32tx+32t-6=3x$$

Add 6 to both sides.

$$8tx^{2}+32tx+32t=3x+6$$

Combine all terms containing t.

$$\left(8x^{2}+32x+32\right)t=3x+6$$

Divide both sides by 8x^{2}+32x+32.

$$\frac{\left(8x^{2}+32x+32\right)t}{8x^{2}+32x+32}=\frac{3x+6}{8x^{2}+32x+32}$$

Dividing by 8x^{2}+32x+32 undoes the multiplication by 8x^{2}+32x+32.

$$t=\frac{3x+6}{8x^{2}+32x+32}$$

Divide 6+3x by 8x^{2}+32x+32.

$$t=\frac{3}{8\left(x+2\right)}$$

Solve for x

$$\left\{\begin{matrix}\\x=-2\text{, }&\text{unconditionally}\\x=-2+\frac{3}{8t}\text{, }&t\neq 0\end{matrix}\right.$$