Find the derivative of each of the following functions, first by using the product rule, then by multiplying each function out and finding the derivative of the higher-order polynomial. Post both solutions along with explanations of the intermediate steps that demonstrate your understanding of the derivative.

1. LaTeX: \left(4x+6\right)\left(9x-5\right)
2. LaTeX: 4x^2\left(3x^3-2x+5x\right)
3. LaTeX: \left(3x^2\:-x\:+1\right)\left(7\:- \:x^6\right)
4. LaTeX: \left(h^3\:-\:5\right)\left(3h^2\:- \:5h\:-\:4\right)

Evaluate 8x2 + 9x − 1 2x3 + 3x2 − 2x dx. solution since the degree of the numerator is less than the degree of the denominator, we don't need to divide. we factor the denominator as 2x3 + 3x2 − 2x = x(2x2 + 3x − 2) = x(2x − 1)(x + 2). since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the form† 8x2 + 9x − 1 x(2x − 1)(x + 2) = correct: your answer is correct. to determine the values of a, b, and c, we multiply both sides of this equation by the product of the denominators, x(2x − 1)(x + 2), obtaining 8x2 + 9x − 1 = a correct: your answer is correct. (x + 2) + bx(x + 2) + cx(2x − 1).