Question

Solve the equation and check your solution. (Some equations have no solution.) $$ (2 x+1)^{2}=4\left(x^{2}+x+1\right) $$

Short answer

The given quadratic equation has no solutions as it simplifies down to 1 = 4, which is a contradiction.

Step-by-step solution

Simplify and rearrange the equation

Start by expanding the square of the binomial and the right side of equation. The equation becomes: \(4x^2 + 4x + 1 = 4x^2 + 4x + 4\). Then, subtract \(4x^2 + 4x\) from both sides to consolidate like terms. This step results in \(4x^2 + 4x + 1 - 4x^2 - 4x = 4x^2 + 4x + 4 - 4x^2 - 4x\). So, \(1 = 4\)

Check for solutions

Since both sides are constants and the left side is not equal to the right side, this equation has no solution.

Verification

There's no need to verify as there is no solution to this equation, as found in step 2.