Solve (-4x-5)^2=(-2x+3)(-2x-3)-2(2x+3)(-3x+5)

$$( - 4 x - 5 ) ^ { 2 } = ( - 2 x + 3 ) ( - 2 x - 3 ) - 2 ( 2 x + 3 ) ( - 3 x + 5 )$$

Solve for x

$$x = -\frac{32}{21} = -1\frac{11}{21} \approx -1.523809524$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-4x-5\right)^{2}.

$$16x^{2}+40x+25=\left(-2x+3\right)\left(-2x-3\right)-2\left(2x+3\right)\left(-3x+5\right)$$

Consider \left(-2x+3\right)\left(-2x-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$16x^{2}+40x+25=\left(-2x\right)^{2}-9-2\left(2x+3\right)\left(-3x+5\right)$$

Expand \left(-2x\right)^{2}.

$$16x^{2}+40x+25=\left(-2\right)^{2}x^{2}-9-2\left(2x+3\right)\left(-3x+5\right)$$

Calculate -2 to the power of 2 and get 4.

$$16x^{2}+40x+25=4x^{2}-9-2\left(2x+3\right)\left(-3x+5\right)$$

Add 2\left(2x+3\right)\left(-3x+5\right) to both sides.

$$16x^{2}+40x+25+2\left(2x+3\right)\left(-3x+5\right)=4x^{2}-9$$

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

$$16x^{2}+40x+25+\left(4x+6\right)\left(-3x+5\right)=4x^{2}-9$$

Use the distributive property to multiply 4x+6 by -3x+5 and combine like terms.

$$16x^{2}+40x+25-12x^{2}+2x+30=4x^{2}-9$$

Combine 16x^{2} and -12x^{2} to get 4x^{2}.

$$4x^{2}+40x+25+2x+30=4x^{2}-9$$

Combine 40x and 2x to get 42x.

$$4x^{2}+42x+25+30=4x^{2}-9$$

Add 25 and 30 to get 55.

$$4x^{2}+42x+55=4x^{2}-9$$

Subtract 4x^{2} from both sides.

$$4x^{2}+42x+55-4x^{2}=-9$$

Combine 4x^{2} and -4x^{2} to get 0.

$$42x+55=-9$$

Subtract 55 from both sides.

$$42x=-9-55$$

Subtract 55 from -9 to get -64.

$$42x=-64$$

Divide both sides by 42.

$$x=\frac{-64}{42}$$

Reduce the fraction \frac{-64}{42} to lowest terms by extracting and canceling out 2.

$$x=-\frac{32}{21}$$