a) \( \sqrt{5 x+4}-1=2 x \)

To solve the equation \( \sqrt{5x + 4} - 1 = 2x \), first isolate the square root. Adding 1 to both sides gives \( \sqrt{5x + 4} = 2x + 1 \). Next, square both sides: \( 5x + 4 = (2x + 1)^2 \), which simplifies to \( 5x + 4 = 4x^2 + 4x + 1 \). Rearranging the equation leads to \( 4x^2 - x - 3 = 0 \). Now, you can factor this or use the quadratic formula to find the roots. The factorization yields \( (4x + 6)(x - 1) = 0 \), resulting in the solutions \( x = \frac{3}{2} \) and \( x = 1 \). Don’t forget to plug these values back into the original equation to check if they hold true!