Question

(dfrac{2x^3 - 5x^2 - 4x - 25}{x - 4})

Explanation:

Step1: Divide leading terms

$\frac{2x^3}{x} = 2x^2$

Step2: Multiply divisor by result

$2x^2(x-4) = 2x^3 - 8x^2$

Step3: Subtract from dividend

$(2x^3 -5x^2) - (2x^3 -8x^2) = 3x^2$

Step4: Bring down next term

New polynomial: $3x^2 -4x$

Step5: Divide leading terms

$\frac{3x^2}{x} = 3x$

Step6: Multiply divisor by result

$3x(x-4) = 3x^2 -12x$

Step7: Subtract from polynomial

$(3x^2 -4x) - (3x^2 -12x) = 8x$

Step8: Bring down last term

New polynomial: $8x -25$

Step9: Divide leading terms

$\frac{8x}{x} = 8$

Step10: Multiply divisor by result

$8(x-4) = 8x -32$

Step11: Subtract to find remainder

$(8x -25) - (8x -32) = 7$

Step12: Combine quotient and remainder

Quotient: $2x^2+3x+8$, Remainder: $7$ → $2x^2 + 3x + 8 + \frac{7}{x-4}$

Answer:

$2x^2 + 3x + 8 + \frac{7}{x-4}$