Answer
The side lengths are 7 and 10.
Work Step by Step
$$x^2+(x+3)^2=149$$
$$x^2+x^2+6x+9=149$$
$$2x^2+6x+9=149$$
$$2x^2+6x-140=0$$
$$x^2+3x-70=0$$
$$(x+10)(x-7)=0$$
$$x=-10,7$$
Length cannot be negative, so x=7.
The side lengths are 7 and 10.