3) Divide \( 2 x^{4}+7 x^{3}-241 x^{2}-192 x+3024 \) by \( x^{2}-5 x-36 \)

To divide \( 2x^{4} + 7x^{3} - 241x^{2} - 192x + 3024 \) by \( x^{2} - 5x - 36 \), we can use polynomial long division. 1. Divide the leading term of the dividend \( 2x^{4} \) by the leading term of the divisor \( x^{2} \) to get \( 2x^{2} \). 2. Multiply the entire divisor \( x^{2} - 5x - 36 \) by \( 2x^{2} \), giving \( 2x^{4} - 10x^{3} - 72x^{2} \). 3. Subtract the result from the original polynomial: \[ (2x^{4} + 7x^{3} - 241x^{2} - 192x + 3024) - (2x^{4} - 10x^{3} - 72x^{2}) = 17x^{3} - 169x^{2} - 192x + 3024 \] 4. Repeat the process by dividing the new leading term \( 17x^{3} \) by \( x^{2} \) to get \( 17x \). 5. Multiply the divisor by \( 17x \): \[ 17x(x^{2} - 5x - 36) = 17x^{3} - 85x^{2} - 612x \] 6. Subtract again: \[ (17x^{3} - 169x^{2} - 192x + 3024) - (17x^{3} - 85x^{2} - 612x) = -84x^{2} + 420x + 3024 \] 7. Next, divide \( -84x^{2} \) by \( x^{2} \) to get \( -84 \). 8. Multiply the divisor by \( -84 \): \[ -84(x^{2} - 5x - 36) = -84x^{2} + 420x + 3024 \] 9. Subtract to find the remainder: \[ (-84x^{2} + 420x + 3024) - (-84x^{2} + 420x + 3024) = 0 \] The result of the division shows that \( 2x^{4} + 7x^{3} - 241x^{2} - 192x + 3024 \) divides perfectly by \( x^{2} - 5x - 36 \). Thus the quotient is: \[ \boxed{2x^{2} + 17x - 84} \]