Solve (4x-8)(3x+3d)+(4x+10)+(3x+5)(7x-23)=540

$$( 4 x - 8 ) ( 3 x + 3 d ) + ( 4 x + 10 ) + ( 3 x + 5 ) ( 7 x - 23 ) = 540$$

Solve for d

$$d=-\frac{11x^{2}-18x-215}{4\left(x-2\right)}$$

$$x\neq 2$$

Steps for Solving Linear Equation

Use the distributive property to multiply 4x-8 by 3x+3d.

$$12x^{2}+12dx-24x-24d+4x+10+\left(3x+5\right)\left(7x-23\right)=540$$

Combine -24x and 4x to get -20x.

$$12x^{2}+12dx-20x-24d+10+\left(3x+5\right)\left(7x-23\right)=540$$

Use the distributive property to multiply 3x+5 by 7x-23 and combine like terms.

$$12x^{2}+12dx-20x-24d+10+21x^{2}-34x-115=540$$

Combine 12x^{2} and 21x^{2} to get 33x^{2}.

$$33x^{2}+12dx-20x-24d+10-34x-115=540$$

Combine -20x and -34x to get -54x.

$$33x^{2}+12dx-54x-24d+10-115=540$$

Subtract 115 from 10 to get -105.

$$33x^{2}+12dx-54x-24d-105=540$$

Subtract 33x^{2} from both sides.

$$12dx-54x-24d-105=540-33x^{2}$$

Add 54x to both sides.

$$12dx-24d-105=540-33x^{2}+54x$$

Add 105 to both sides.

$$12dx-24d=540-33x^{2}+54x+105$$

Add 540 and 105 to get 645.

$$12dx-24d=645-33x^{2}+54x$$

Combine all terms containing d.

$$\left(12x-24\right)d=645-33x^{2}+54x$$

The equation is in standard form.

$$\left(12x-24\right)d=645+54x-33x^{2}$$

Divide both sides by 12x-24.

$$\frac{\left(12x-24\right)d}{12x-24}=\frac{645+54x-33x^{2}}{12x-24}$$

Dividing by 12x-24 undoes the multiplication by 12x-24.

$$d=\frac{645+54x-33x^{2}}{12x-24}$$

Divide 645-33x^{2}+54x by 12x-24.

$$d=\frac{215+18x-11x^{2}}{4\left(x-2\right)}$$

Solve for x

$$x=\frac{\sqrt{4d^{2}+52d+2446}-2d+9}{11}$$

$$x=\frac{-\sqrt{4d^{2}+52d+2446}-2d+9}{11}$$