Solving the Polynomial Division: (x² + x  17) ÷ (x  4)
This article will guide you through the process of dividing the polynomial (x² + x  17) by (x  4). We will use the method of long division to solve this problem.
Long Division Steps

Set up the division: Write the dividend (x² + x  17) inside the division symbol and the divisor (x  4) outside.
_______ x  4  x² + x  17

Divide the leading terms: Divide the leading term of the dividend (x²) by the leading term of the divisor (x). This gives us x. Write this quotient above the x² term in the dividend.
x x  4  x² + x  17

Multiply the quotient by the divisor: Multiply the quotient (x) by the entire divisor (x  4) and write the result below the dividend.
x x  4  x² + x  17 x²  4x

Subtract: Subtract the result from the dividend. Be sure to distribute the negative sign.
x x  4  x² + x  17 x²  4x  5x  17

Bring down the next term: Bring down the next term (17) from the dividend.
x x  4  x² + x  17 x²  4x  5x  17

Repeat steps 25: Repeat the process by dividing the leading term of the new dividend (5x) by the leading term of the divisor (x). This gives us 5. Write this quotient above the 17.
x + 5 x  4  x² + x  17 x²  4x  5x  17 5x  20

Subtract: Subtract the result from the new dividend.
x + 5 x  4  x² + x  17 x²  4x  5x  17 5x  20  3

The remainder: The final result is the remainder (3).
Final Solution
Therefore, (x² + x  17) ÷ (x  4) = x + 5 + 3/(x  4)