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The easiest way to do this is to square both sides but you have to check the answers that you get because some may be invalid.

Squaring both sides as melody said. we get: 

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

(4x+3)^2=(4x+3)(4x+3)=16x^2+12x+12x+9=16x^2+24x+9 

So we have

\(4x^2+4x+1=16x^2+24x+9\)

Subtract 1 

\(4x^2+4x=16x^2+24x+8\)

subtract 4x 

\(4x^2=16x^2+20x+8\)

Rearrange 

\(16x^2+20x+8=4x^2\)

Subtract 4x^2

\(12x^2+20x+8=0\)

Factor 4

\(4(3x^2+5x+2)\)=0

\(3x^2+5x+2=0\)

x=\(\frac{-2}{3}\) or -1 

Subsituite back and check 

2(-1)+1=4(-1)+3 

-1 = -1 correct! 

-1 is a solution 

Subsituite -2/3 and check, Correct!