To find the remainder of the division of the polynomial x^4 + 4x^3 + 2x^2 + x + 4 by x^2 + 3x, we can use polynomial long division.
1. First, divide the leading term of the dividend (x^4) by the leading term of the divisor (x^2), which gives us x^2.
2. Next, multiply the entire divisor by x^2: x^2(x^2 + 3x) = x^4 + 3x^3.
3. Subtract this result from the original polynomial: (x^4 + 4x^3 + 2x^2 + x + 4) – (x^4 + 3x^3) = (4x^3 – 3x^3) + 2x^2 + x + 4 = x^3 + 2x^2 + x + 4.
4. Now, repeat the process: divide x^3 by x^2, which gives us x.
5. Multiply the entire divisor by x: x(x^2 + 3x) = x^3 + 3x^2.
6. Subtract again: (x^3 + 2x^2 + x + 4) – (x^3 + 3x^2) = (2x^2 – 3x^2) + x + 4 = -x^2 + x + 4.
7. Finally, divide -x^2 by x^2, which gives us -1.
8. Multiply the divisor by -1: -1(x^2 + 3x) = -x^2 – 3x.
9. Subtract once more: (-x^2 + x + 4) – (-x^2 – 3x) = (x + 3x) + 4 = 4x + 4.
At this point, the degree of the remaining polynomial 4x + 4 is less than the degree of the divisor x^2 + 3x. Therefore, 4x + 4 is the remainder.
Remainder: 4x + 4