To divide the polynomials 4x^4 + 4x - 10 by 2x^2 - 3, follow these steps:
Step 1: Arrange the polynomials in descending order of exponents:
4x^4 + 0x^3 + 0x^2 + 4x - 10
2x^2 + 0x - 3
Step 2: Divide the first term of the dividend (4x^4) by the first term of the divisor (2x^2), which gives us 2x^2.
Step 3: Multiply the divisor (2x^2) by the result from step 2 (2x^2), which gives us 4x^4.
Step 4: Subtract the result from step 3 (4x^4) from the dividend (4x^4 + 4x - 10), which cancels out the first term. The result is:
0x^4 + 4x - 10
Step 5: Bring down the next term from the dividend (4x) to form a new dividend:
0x^4 + 4x - 10
2x^2 - 3
Step 6: Divide the first term of the new dividend (4x) by the first term of the divisor (2x^2), which gives us 2x.
Step 7: Multiply the divisor (2x^2) by the result from step 6 (2x), which gives us 4x^2.
Step 8: Subtract the result from step 7 (4x^2) from the new dividend (4x^2 + 4x - 10), which cancels out the first term. The result is:
0x^2 + 4x - 10
Step 9: Bring down the next term from the new dividend (-10) to form a new dividend:
0x^2 + 4x - 10
2x^2 - 3
Step 10: Divide the first term of the new dividend (-10) by the first term of the divisor (2x^2), which gives us -5.
Step 11: Multiply the divisor (2x^2) by the result from step 10 (-5), which gives us -10x^2.
Step 12: Subtract the result from step 11 (-10x^2) from the new dividend (0x^2 + 4x - 10), which cancels out the first term. The result is:
0x^2 + 4x - 10 + (-10x^2) = -10x^2 + 4x - 10.
Step 13: We have no more terms to bring down from the dividend, so our division is complete.
Therefore, the quotient is 2x + (-5) and the remainder is -10x^2 + 4x - 10.