Solve 4(x+1)(x-2)=-(1+2x)^2

$$4 ( x + 1 ) ( x - 2 ) = - ( 1 + 2 x ) ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{14}}{4}\approx 0.935414347$$

$$x=-\frac{\sqrt{14}}{4}\approx -0.935414347$$

Steps by Finding Square Root

Use the distributive property to multiply 4 by x+1.

$$\left(4x+4\right)\left(x-2\right)=-\left(1+2x\right)^{2}$$

Use the distributive property to multiply 4x+4 by x-2 and combine like terms.

$$4x^{2}-4x-8=-\left(1+2x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+2x\right)^{2}.

$$4x^{2}-4x-8=-\left(1+4x+4x^{2}\right)$$

To find the opposite of 1+4x+4x^{2}, find the opposite of each term.

$$4x^{2}-4x-8=-1-4x-4x^{2}$$

Add 4x to both sides.

$$4x^{2}-4x-8+4x=-1-4x^{2}$$

Combine -4x and 4x to get 0.

$$4x^{2}-8=-1-4x^{2}$$

Add 4x^{2} to both sides.

$$4x^{2}-8+4x^{2}=-1$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-8=-1$$

Add 8 to both sides.

$$8x^{2}=-1+8$$

Add -1 and 8 to get 7.

$$8x^{2}=7$$

Divide both sides by 8.

$$x^{2}=\frac{7}{8}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{14}}{4} x=-\frac{\sqrt{14}}{4}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by x+1.

$$\left(4x+4\right)\left(x-2\right)=-\left(1+2x\right)^{2}$$

Use the distributive property to multiply 4x+4 by x-2 and combine like terms.

$$4x^{2}-4x-8=-\left(1+2x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+2x\right)^{2}.

$$4x^{2}-4x-8=-\left(1+4x+4x^{2}\right)$$

To find the opposite of 1+4x+4x^{2}, find the opposite of each term.

$$4x^{2}-4x-8=-1-4x-4x^{2}$$

Subtract -1 from both sides.

$$4x^{2}-4x-8-\left(-1\right)=-4x-4x^{2}$$

The opposite of -1 is 1.

$$4x^{2}-4x-8+1=-4x-4x^{2}$$

Add 4x to both sides.

$$4x^{2}-4x-8+1+4x=-4x^{2}$$

Add -8 and 1 to get -7.

$$4x^{2}-4x-7+4x=-4x^{2}$$

Combine -4x and 4x to get 0.

$$4x^{2}-7=-4x^{2}$$

Add 4x^{2} to both sides.

$$4x^{2}-7+4x^{2}=0$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 0 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 8\left(-7\right)}}{2\times 8}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 8\left(-7\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{0±\sqrt{-32\left(-7\right)}}{2\times 8}$$

Multiply -32 times -7.

$$x=\frac{0±\sqrt{224}}{2\times 8}$$

Take the square root of 224.

$$x=\frac{0±4\sqrt{14}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{0±4\sqrt{14}}{16}$$

Now solve the equation x=\frac{0±4\sqrt{14}}{16} when ± is plus.

$$x=\frac{\sqrt{14}}{4}$$

Now solve the equation x=\frac{0±4\sqrt{14}}{16} when ± is minus.

$$x=-\frac{\sqrt{14}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{14}}{4} x=-\frac{\sqrt{14}}{4}$$