Solve (A+B)x^2+(B-5)x+A-57=(L)^2

$$( A + B ) x ^ { 2 } + ( B - 5 ) x + A - 57 = ( L ) ^ { 2 }$$

Solve for A

$$A=-\frac{Bx^{2}+Bx-5x-L^{2}-57}{x^{2}+1}$$

Steps for Solving Linear Equation

Use the distributive property to multiply A+B by x^{2}.

$$Ax^{2}+Bx^{2}+\left(B-5\right)x+A-57=L^{2}$$

Use the distributive property to multiply B-5 by x.

$$Ax^{2}+Bx^{2}+Bx-5x+A-57=L^{2}$$

Subtract Bx^{2} from both sides.

$$Ax^{2}+Bx-5x+A-57=L^{2}-Bx^{2}$$

Subtract Bx from both sides.

$$Ax^{2}-5x+A-57=L^{2}-Bx^{2}-Bx$$

Add 5x to both sides.

$$Ax^{2}+A-57=L^{2}-Bx^{2}-Bx+5x$$

Add 57 to both sides.

$$Ax^{2}+A=L^{2}-Bx^{2}-Bx+5x+57$$

Reorder the terms.

$$Ax^{2}+A=-Bx^{2}-Bx+5x+L^{2}+57$$

Combine all terms containing A.

$$\left(x^{2}+1\right)A=-Bx^{2}-Bx+5x+L^{2}+57$$

The equation is in standard form.

$$\left(x^{2}+1\right)A=57+L^{2}+5x-Bx-Bx^{2}$$

Divide both sides by x^{2}+1.

$$\frac{\left(x^{2}+1\right)A}{x^{2}+1}=\frac{57+L^{2}+5x-Bx-Bx^{2}}{x^{2}+1}$$

Dividing by x^{2}+1 undoes the multiplication by x^{2}+1.

$$A=\frac{57+L^{2}+5x-Bx-Bx^{2}}{x^{2}+1}$$

Solve for B

$$\left\{\begin{matrix}B=-\frac{Ax^{2}-5x-L^{2}+A-57}{x\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 0\\B\in \mathrm{R}\text{, }&\left(A=L^{2}+57\text{ and }x=0\right)\text{ or }\left(A=\frac{L^{2}}{2}+26\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply A+B by x^{2}.

$$Ax^{2}+Bx^{2}+\left(B-5\right)x+A-57=L^{2}$$

Use the distributive property to multiply B-5 by x.

$$Ax^{2}+Bx^{2}+Bx-5x+A-57=L^{2}$$

Subtract Ax^{2} from both sides.

$$Bx^{2}+Bx-5x+A-57=L^{2}-Ax^{2}$$

Add 5x to both sides.

$$Bx^{2}+Bx+A-57=L^{2}-Ax^{2}+5x$$

Subtract A from both sides.

$$Bx^{2}+Bx-57=L^{2}-Ax^{2}+5x-A$$

Add 57 to both sides.

$$Bx^{2}+Bx=L^{2}-Ax^{2}+5x-A+57$$

Reorder the terms.

$$Bx^{2}+Bx=-Ax^{2}+5x+L^{2}-A+57$$

Combine all terms containing B.

$$\left(x^{2}+x\right)B=-Ax^{2}+5x+L^{2}-A+57$$

The equation is in standard form.

$$\left(x^{2}+x\right)B=57-A+L^{2}+5x-Ax^{2}$$

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

$$\frac{\left(x^{2}+x\right)B}{x^{2}+x}=\frac{57-A+L^{2}+5x-Ax^{2}}{x^{2}+x}$$

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

$$B=\frac{57-A+L^{2}+5x-Ax^{2}}{x^{2}+x}$$

Divide -Ax^{2}+5x+L^{2}-A+57 by x^{2}+x.

$$B=\frac{57-A+L^{2}+5x-Ax^{2}}{x\left(x+1\right)}$$