What is the result of dividing \( 6x^4 + 11x^2 - 10 \) by \( 2x^2 + 5 \)?

To divide the polynomial ( 6x^4 + 11x^2 - 10 ) by ( 2x^2 + 5 ), we can utilize polynomial long division.

First, we write down the division setup as we would with numerical long division. We start with the leading term of the dividend, ( 6x^4 ), divided by the leading term of the divisor ( 2x^2 ). This gives us ( 3x^2 ), which we place as the first term of our quotient.

Next, we multiply ( 3x^2 ) by ( 2x^2 + 5 ), resulting in ( 6x^4 + 15x^2 ). We subtract this product from the original polynomial:

[

(6x^4 + 11x^2 - 10) - (6x^4 + 15x^2) = 11x^2 - 15x^2 - 10 = -4x^2 - 10.

]

Now, we repeat the division process using the leading term ( -4x^2 ). We divide this by ( 2x^2 ),