Simplify by completely dividing the following expression. (a⁴ + 13a² + 5a + 12)/(a² + 6)

Find the polynomial, if 3x² + 11x + 4 yields a quotient of 3x - 1 and a remainder of 8 when divided by this polynomial.

Work out the value of k that makes 9x + 1 a factor of 18x³ + 74x² - 145x + k.

A rectangle has an area of 0.7x³ - 0.695x² - 0.02x + 0.015 and a length of x + 0.15 units. Solve for the polynomial that describes its width.