To find x, first we'll split the triangle in two right triangles.
The first triangle will be the one with sides (8,x+4,y) and the second triangle will be the one with sides (12,2x+1,y).
As you can see, these two triangles share the side 'y'.
To solve we'll do the following; first solve for 'y' by applying the Pythagorean theorem in the second triangle, and finally solve for 'x' applying Pythagorean theorem in the first triangle.
From the second triangle (12,2x+1,y) we solve for 'y' by writing:
[tex] y^{2}=12^{2}- (2x+1)^{2}=144-4 x^{2} -4x-1=[/tex]
Now that we know 'y^2' we apply Pythagorean Theorem in the first triangle (8,x+4,y)
[tex] 8^{2}= (x+4)^{2}+ y^{2}= (x^{2}+8x+16)+(143-4 x^{2} -4x)[/tex]
[tex]64=159-3 x^{2} +4x[/tex]
Now you solve the quadratic equation, you'll get a positive and negative value; as this are lengths, you take the positive and get:
[tex]x= \frac{19}{3} [/tex]
