calculus - Evaluating a limit using only rationalization and algebraic methods

Evaluate (without using de L'Hôpital Rule)

$$\lim_{x \to -3}\frac{\sqrt{3 x^2+10 x+4}-\sqrt{x^2+3 x+1}}{\sqrt{5 x^2+11 x+5}-\sqrt{2 x^2+x+2}} = \frac 58 \sqrt{17}$$

I have to evaulate the limit of this function as it approaches $-3$. I have tried plugging it in but I get $0/0$. Then I tried LCD, but the $\sqrt x-\sqrt y$ does not equal $\sqrt{x-y}$. Then I tried multiplying the conjugate of the denominator but after all of that I still get $0/0$. I cannot use the calculator except to check my answers after completion. However when I checked there is a limit at $-3$ so I resorted to looking online and it gave me the answer $\frac58\sqrt{17}$, but I cannot seem to get this answer without using L'Hôpital's rule.

Answer

Hint 1) $\frac{\sqrt{a}- \sqrt{b}}{\sqrt{c}-\sqrt{d}} = \frac{(a-b)(\sqrt{c} + \sqrt{d})}{(c+d)(\sqrt{a}+\sqrt{b})}$

Hint 2) $2x^2 + 7x +3 = (2x+1)(x+3)$, this is $a+b$

Hint 3) Try hint 2) with $c+d$.

What do you get?