Solve D=(2x+3)^2+(2x+3)(7x-2)

$$D = ( 2 x + 3 ) ^ { 2 } + ( 2 x + 3 ) ( 7 x - 2 )$$

Solve for D

$$D=\left(2x+3\right)\left(9x+1\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$D=4x^{2}+12x+9+\left(2x+3\right)\left(7x-2\right)$$

Use the distributive property to multiply 2x+3 by 7x-2 and combine like terms.

$$D=4x^{2}+12x+9+14x^{2}+17x-6$$

Combine 4x^{2} and 14x^{2} to get 18x^{2}.

$$D=18x^{2}+12x+9+17x-6$$

Combine 12x and 17x to get 29x.

$$D=18x^{2}+29x+9-6$$

Subtract 6 from 9 to get 3.

$$D=18x^{2}+29x+3$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{72D+625}-29}{36}$$

$$x=\frac{\sqrt{72D+625}-29}{36}$$

Solve for x

$$x=\frac{-\sqrt{72D+625}-29}{36}$$

$$x=\frac{\sqrt{72D+625}-29}{36}\text{, }D\geq -\frac{625}{72}$$