Use polynomial long division to perform the indi-cated division. Write the polynomial in the form p(x) = d(x)q(x) + r(x).(a) (5x4 − 3x3 + 2x2 − 1) ÷ (x2 + 4)(b) (−x5 + 7x3 − x) ÷ (x3 − x2 + 1)(c) (9x3 + 5) ÷ (2x − 3)(d) (4x2 − x − 23) ÷ (x2 − 1)

Sure, let's solve these step by step:

(a) (5x^4 − 3x^3 + 2x^2 − 1) ÷ (x^2 + 4)

First, divide the first term in the numerator by the first term in the denominator to get the first term of the quotient, 5x^2. Multiply the denominator by 5x^2 and subtract the result from the original polynomial to get the new numerator, -3x^3. Repeat this process until you can't divide anymore. The final quotient is 5x^2 and the remainder is -3x^3 + 2x^2 - 1. So, p(x) = d(x)q(x) + r(x) becomes 5x^4 - 3x^3 + 2x^2 - 1 = (x^2 + 4)(5x^2) - 3x^3 + 2x^2 - 1.

(b) (−x^5 + 7x^3 − x) ÷ (x^3 − x^2 + 1)

The first term of the quotient is -x^2. The new numerator is 7x^3 - x. Repeat the process to get the final quotient of -x^2 + x and the remainder of 0. So, p(x) = d(x)q(x) + r(x) becomes -x^5 + 7x^3 - x = (x^3 - x^2 + 1)(-x^2 + x).

(c) (9x^3 + 5) ÷ (2x − 3)

The first term of the quotient is 4.5x^2. The new numerator is 5. The final quotient is 4.5x^2 and the remainder is 5. So, p(x) = d(x)q(x) + r(x) becomes 9x^3 + 5 = (2x - 3)(4.5x^2) + 5.

(d) (4x^2 − x − 23) ÷ (x^2 − 1)

The first term of the quotient is 4. The new numerator is -x - 23. The final quotient is 4 and the remainder is -x - 23. So, p(x) = d(x)q(x) + r(x) becomes 4x^2 - x - 23 = (x^2 - 1)(4) - x - 23.