What is the remainder when dividing the polynomial 4x^3 + 5x^2 + 3x + 1 by x^2?

To find the remainder when dividing the polynomial 4x³ + 5x² + 3x + 1 by x², we can use polynomial long division or the Remainder Theorem.

Since we are dividing by a polynomial of degree 2, the remainder will be a polynomial of degree less than 2. Therefore, the remainder can be expressed in the form:

R(x) = Ax + B

where A and B are constants that we need to determine.

We will perform the polynomial long division:

1. Divide the leading term of the dividend (4x³) by the leading term of the divisor (x²): 4x³ ÷ x² = 4x.
2. Multiply the entire divisor x² by the result (4x): 4x * x² = 4x³.
3. Subtract this result from the original polynomial:

(4x3 + 5x2 + 3x + 1) - (4x3) = 5x2 + 3x + 1.

Next, repeat the process:

1. Divide the leading term of the new polynomial (5x²) by the leading term of the divisor (x²): 5x² ÷ x² = 5.
2. Multiply the entire divisor by this result (5): 5 * x² = 5x².
3. Subtract again:

(5x2 + 3x + 1) - (5x2) = 3x + 1.

Now, we cannot divide 3x + 1 by x² since the degree of the remainder (1) is less than the degree of the divisor (2).

Thus, the remainder when dividing 4x³ + 5x² + 3x + 1 by x² is:

R(x) = 3x + 1.

In conclusion, the final answer is:

R(x) = 3x + 1.