We can add $x$ and square each term, following which the sequence is a quadratic polynomial in $x$.
$378.0872915, 496.4625614, 591.2874514, 672.6574576, 745.0246992\dots$
Adding $x$, we get,
$378.0872915, 497.4625614, 593.2874514, 675.6574576, 749.0246992\dots$
Squaring, we get,
$142950, 247469, 351990, 456513, 561038\dots$
It is not difficult to solve systems of equations or use Lagrange interpolation to get the quadratic:
$x^2+104518x+142950$
For a term to be an integer, we need to solve the Diophantine equation:
$x^2+104518x+142950=y^2$
To reduce the number of terms, let's try to complete the square:
$(x+52259)^2-2730860131=y^2$ or $y^2-(x+52259)^2=-2730860131$
By factoring $-2730860131$ into $2$ factors of the same parity, we can find integer solutions to this, the $2$ factors being $y+(x+52259)$ and $y-(x+52259)$.
If the $x$ coefficient is odd, say the equation was,
$x^2+104517x+142950=y^2$
We can complete the square again:
$(x+52258.5)^2-2730807872.25=y^2$ or $y^2-(x+52258.5)^2=-2730807872.25$
Multiplying both sides by $4$, we get:
$(2y)^2-(2x+104517)^2=-10923231489$
We need to find $2$ odd factors of $-10923231489$ to be $2y+(2x+104517)$ and $2y-(2x+104517)$.
To answer your question if it is possible to do without factoring, suppose you find the solution with an alternative method. Then we can take out the quadratic and plug $x$ in.
As such, we can find $y$ by taking the square root.
As such, we would have found the factors of some big number, which is expressed in terms of $x$ and $y$ ($y+(x+52259)$ and $y-(x+52259)$ or $2y+(2x+104517)$ and $2y-(2x+104517)$), which are both obtainable. So, if you find the solution, it means you have somehow factored the number.
Best Answer
Nowadays, the #1 method for predicting the next number from a sequence (assuming the sequence has come up in a "natural" way) is to look it up in the Online Encyclopedia of Integer Sequences. In his 1973 book, A Handbook of Integer Sequences, Sloane gives some suggestions as to what to do if your sequence is not in the Encyclopedia/Handbook. These include,
Add or subtract 1 or 2 from all the terms, and try looking it up again;
Multiply all the terms by 2, or divide by any common factor, and try looking it up again;
Look for a recurrence.
Sloane elaborates on this last suggestion. He mentions the method of differences, where you replace the sequence $a_0,a_1,\dots$ with $a_1-a_0,a_2-a_1,\dots$ and, if necessary, repeat the differencing, until you get something with an obvious pattern. Of course, then you have to know what to do with a recurrence once you have one, but that's another story.
Sloane also says that if a sequence is close to a known sequence, you can try subtracting off the known sequence, and then dealing with the residual by one of the above methods.
If the ratios $a_{n+1}/a_n$ seem to be close to a recognizable sequence $r_n$, then look at the sequence given by $a_{n+1}-r_na_n$.
Factoring the numbers in a sequence, or in a sequence close to the given sequence, will often give a clue as to what is going on.
For examples of all these principles (and others that I haven't mentioned) in operation, I refer you to the Handbook.