$$ 5 \cdot \sqrt{3x+4} + 8 = 6\cdot \sqrt{5x+1} -3 \quad\vert +3$$
$$ 5 \cdot \sqrt{3x+4} + 11 = 6\cdot \sqrt{5x+1} \quad\vert -5 \cdot \sqrt{3x+4} $$
$$ 11 = 6\cdot \sqrt{5x+1} \quad -5 \cdot \sqrt{3x+4} $$
$$ 11^2 = (6\cdot \sqrt{5x+1} \quad -5 \cdot \sqrt{3x+4})^2 $$
$$ 121 = 36 \cdot( 5x+1)-2 \cdot 6\cdot \sqrt{5x+1} \cdot 5 \cdot \sqrt{3x+4} +25 \cdot (3x+4)$$
$$ 121 = 180x+36- 60\cdot \sqrt{5x+1} \cdot \sqrt{3x+4} +75x+100$$
$$ 121 = 255x+136- 60\cdot \sqrt{(5x+1) \cdot (3x+4)} $$
$$ 0 = 255x+15- 60\cdot \sqrt{15x^2+3x+20x +4} $$
$$ 60\cdot \sqrt{15x^2+3x+20x +4} = 255x+15 \quad\vert :15$$
$$ 4\cdot \sqrt{15x^2+23x +4} = 17x+1$$
$$ ( 4\cdot \sqrt{15x^2+23x +4} )^2 = (17x+1)^2$$
$$ 16\cdot (15x^2+23x +4 ) =17^2 x^2 +2 \cdot 17x+1$$
$$ 240 x^2+368x +64 =289 x^2 +34x+1$$
$$ 368x +64 =49 x^2 +34x+1$$
$$ 64 =49 x^2 -334x+1$$
$$ 0 =49 x^2 -334x-63$$
$$x_1=7\quad ;\quad x_2= -\frac{9}{49}$$
