Final answer:
Option c, 2x² + 9x - 2, is the correct choice as it is a polynomial with a quadratic and a linear term when simplified; but it is not technically a 'quadratic binomial' since it includes a linear term as well.
Step-by-step explanation:
The question asks which of the following is a quadratic binomial when simplified. A quadratic binomial is of the form ax² + bx or ax² + c, where a, b, and c are constants and a is not zero. Among the given options, we look for an expression that, when simplified, has the exponents of the variable to be 2 and 0, which would make it a binomial and quadratic if there are two terms.
Option a simplifies to 5 - 2x² + 52, which is not a binomial since it has three terms. Option b simplifies to -10 + 4x + 40x² - 823, which is not a binomial as it has more than two terms, and option d, 142 + 3.22, does not contain a variable term. Therefore, option c, 2x² + 9x - 2, is the correct answer because it is a binomial with a quadratic term and a linear term; however, it is not a quadratic binomial, as it includes the linear term as well.