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)

Sure, let's perform the polynomial long division step by step.

Step 1: Write the division in long division format.

         _______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1

Step 2: Divide the first term of the dividend (5x⁴) by the first term of the divisor (x²) to get the first term of the quotient (5x²). Write this above the line.

         5x²
         _______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1

Step 3: Multiply the divisor (x² + 4) by the first term of the quotient (5x²) and subtract the result from the dividend.

         5x²
         _______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
       - (5x⁴ + 20x²)
       -----------------
             -3x³ - 18x² - 1

Step 4: Repeat the process. Divide the first term of the new dividend (-3x³) by the first term of the divisor (x²) to get the next term of the quotient (-3x). Write this above the line.

         5x² - 3x
         _______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
       - (5x⁴ + 20x²)
       -----------------
             -3x³ - 18x² - 1
       - (-3x³ - 12x)
       -----------------
             -6x² + 12x - 1

Step 5: Repeat the process again. Divide the first term of the new dividend (-6x²) by the first term of the divisor (x²) to get the next term of the quotient (-6). Write this above the line.

         5x² - 3x - 6
         _______________
x² + 4 | 5x⁴ - 3x³ + 2x² - 1
       - (5x⁴ + 20x²)
       -----------------
             -3x³ - 18x² - 1
       - (-3x³ - 12x)
       -----------------
             -6x² + 12x - 1
       - (-6x² - 24)
       -----------------
             12x + 23

Step 6: Since the degree of the new dividend (12x + 23) is less than the degree of the divisor (x² + 4), we stop here. The new dividend is the remainder.

So, the polynomial in the form p(x) = d(x)q(x) + r(x) is:

5x⁴ - 3x³ + 2x² - 1 = (x² + 4)(5x² - 3x - 6) + (12x + 23)