No need to use `FindInstance`. In fact you can generate all possible equations with prescribed roots `r1` and `r2`.

    dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3];
    r1r2 = RandomSample[Range[-10, 10], 2]
    
    (d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
        k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
     Thread[{r1, r2} -> r1r2]
    
    Solve[%]
    
    Clear[dtk, r1r2]

---

    {-5, 4}
    
    129 + 108 x + 60 x^2 == (7 + 8 x)^2
    
    {{x -> -5}, {x -> 4}}

**Update:**

    Table[dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3]; 
      r1r2 = RandomSample[Range[-10, 10], 
        2]; (d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
          k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
       Thread[{r1, r2} -> r1r2], 100];

    Grid@Partition[
      Select[Sqrt[#[[1]]] == (#[[2]] /. x__^2 :> x) & /@ %, 
       Length@Solve[#] == 2 &], 2]
    Clear[dtk, r1r2]
---

$\begin{array}{ll}
 \sqrt{75 x^2+84 x-27}=9 x+3 & \sqrt{6 x^2-4 x+103}=7-2 x \\
 \sqrt{61 x^2-40 x+1}=1-8 x & \sqrt{42 x^2+24 x+478}=6 x+10 \\
 \sqrt{x^2-100 x-316}=-3 x-2 & \sqrt{5 x^2-26 x+148}=10-2 x \\
 \sqrt{74 x^2+250 x+745}=-8 x-5 & \sqrt{4 x^2-x-5}=x+1 \\
 \sqrt{97 x^2-164 x+28}=7-10 x & \sqrt{9 x^2+64 x-104}=4 x+1 \\
 \sqrt{40 x^2-119 x-38}=4-7 x & \sqrt{21 x^2-58 x+25}=5-5 x \\
 \sqrt{62 x^2+142 x+72}=-8 x-10 & \sqrt{50 x^2-99 x+86}=7 x-6 \\
 \sqrt{79 x^2-38 x-124}=9 x-4 & \sqrt{52 x^2+131 x+79}=-7 x-7 \\
 \sqrt{105 x^2+200 x+99}=-10 x-8 & \sqrt{29 x^2+111 x-82}=6 x+4 \\
 \sqrt{76 x^2+101 x-176}=9 x+2 & \sqrt{2 x^2-30 x-56}=2-2 x \\
 \sqrt{59 x^2-230 x+300}=7 x-10 & \sqrt{83 x^2+64 x+109}=9 x+5 \\
 \sqrt{97 x^2-98 x+16}=4-10 x & \sqrt{29 x^2+60 x+32}=5 x+6 \\
\end{array}$