Final answer:
In dividing the polynomial (8x² + 70x + 47) by (4x + 5), we identify the dividend, divisor, quotient, and remainder using long division, and verify the results using the division algorithm to ensure that the initial polynomial is equivalent to the divisor times the quotient plus the remainder.
Step-by-step explanation:
When dividing polynomials, the first step is to set up the long division in the same way you might with numbers. Here, your dividend is (8x² + 70x + 47) and your divisor is (4x + 5). You begin the division by asking how many times does 4x go into 8x², which is 2x. You multiply this quotient with the divisor and subtract the result from the dividend. Repeat this process until you cannot divide anymore, and you will be left with a remainder if the divisor does not go evenly into the dividend.
Following these steps, you should be able to obtain the quotient and the remainder for the expression given. Remember to double-check your work using the division algorithm, which states that the dividend equals the divisor times the quotient plus the remainder. Mathematically, this is d = (qd) + r, where d is the dividend, qd is the quotient times the divisor, and r is the remainder.
To ensure the result is reasonable, eliminate terms wherever possible to simplify the algebra, and utilize your calculator to check the arithmetic involved, expressing results in proper scientific notation if necessary.