To divide the polynomial (x^4 - x^3 + x^2 + 3x) by (x^2 - 2x + 3), we can use long division.

First, we divide the highest degree term of the dividend (x^4) by the highest degree term of the divisor (x^2). This gives us x^2 as the first term of the quotient.

Next, we multiply the entire divisor (x^2 - 2x + 3) by the first term of the quotient (x^2). This gives us (x^4 - 2x^3 + 3x^2).

Now, we subtract this product from the dividend (x^4 - x^3 + x^2 + 3x) to get the new dividend:

(x^4 - x^3 + x^2 + 3x) - (x^4 - 2x^3 + 3x^2) = (x^4 - x^3 + x^2 + 3x - x^4 + 2x^3 - 3x^2) = (-x^3 + 4x^3 - 2x^2 + 3x^2 + 3x) = (3x^3 + x^2 + 3x)

Now, we repeat the process with the new dividend (3x^3 + x^2 + 3x) and the divisor (x^2 - 2x + 3).

We divide the highest degree term of the new dividend (3x^3) by the highest degree term of the divisor (x^2). This gives us 3x as the next term of the quotient.

Next, we multiply the entire divisor (x^2 - 2x + 3) by the next term of the quotient (3x). This gives us (3x^3 - 6x^2 + 9x).

We subtract this product from the new dividend (3x^3 + x^2 + 3x) to get the new dividend:

(3x^3 + x^2 + 3x) - (3x^3 - 6x^2 + 9x) = (3x^3 + x^2 + 3x - 3x^3 + 6x^2 - 9x) = (7x^2 - 6x)

Now, we repeat the process with the new dividend (7x^2 - 6x) and the divisor (x^2 - 2x + 3).

We divide the highest degree term of the new dividend (7x^2) by the highest degree term of the divisor (x^2). This gives us 7 as the next term of the quotient.

Next, we multiply the entire divisor (x^2 - 2x + 3) by the next term of the quotient (7). This gives us (7x^2 - 14x + 21).

We subtract this product from the new dividend (7x^2 - 6x) to get the remainder:

(7x^2 - 6x) - (7x^2 - 14x + 21) = (7x^2 - 6x - 7x^2 + 14x - 21) = (8x - 21)

Therefore, the quotient is x^2 + 3x + 7 and the remainder is 8x - 21.