Step-by-Step Solution

Solve the equation: $3x^2 + 15x - 2\sqrt{x^2 + 5x + 1} = 2$

Step-by-Step Solution

Step 1: Express outside terms in terms of the radicand

Rewrite $3x^2 + 15x$: $3x^2 + 15x = 3(x^2 + 5x + 1) - 3$

Substitute into the equation: $3(x^2 + 5x + 1) - 3 - 2\sqrt{x^2 + 5x + 1} = 2$

$3(x^2 + 5x + 1) - 2\sqrt{x^2 + 5x + 1} - 5 = 0$

Step 2: Substitute radical variable $u$

Let $u = \sqrt{x^2 + 5x + 1} \ge 0$: $3u^2 - 2u - 5 = 0$

$(3u - 5)(u + 1) = 0$

$u = \frac{5}{3} \quad \text{or} \quad u = -1 \quad (\text{rejected as } u \ge 0)$

Step 3: Solve for $x$ with $u = \frac{5}{3}$

$x^2 + 5x + 1 = \left(\frac{5}{3}\right)^2 = \frac{25}{9}$

$9(x^2 + 5x + 1) = 25$

$9x^2 + 45x - 16 = 0$

$(3x - 1)(3x + 16) = 0$

$x = \frac{1}{3} \quad \text{or} \quad x = -\frac{16}{3}$

Step 4: Check solutions

Both candidates yield radicand $\dfrac{25}{9}$ and satisfy the original equation $\checkmark$.

$\boxed{\text{Solution Set: } \left\{\frac{1}{3}, \; -\frac{16}{3}\right\}}$