Final answer:
The solution to the inequality x² + 36 > 12x is all real numbers except x = 6, matching option C. The second inequality x² + 2x + 8 ≤ 0 has no solution since the expression is always positive. The correct answer is option c.
Step-by-step explanation:
To solve the inequality x² + 36 > 12x, we first bring all terms to one side of the inequality to get a quadratic inequality of the form ax² + bx + c > 0. We do this by subtracting 12x from both sides, resulting in x² - 12x + 36 > 0. This can be factored into (x - 6)² > 0. Since a square is always non-negative, and we want it to be strictly greater than zero, we find that x cannot be equal to 6. Thus, the solution is all real numbers except x = 6, which can be represented as x | x ∈ R and x ≠ 6, corresponding to option C.
The second part of the question seems to be asking for the solution to a quadratic inequality of the form x² + 2x + 8 ≤ 0. However, there is a fundamental issue; the quadratic expression x² + 2x + 8 has no real roots as its discriminant b² - 4ac is negative (2² - 4*1*8 = -28). As such, the parabola described by the quadratic expression does not cross the x-axis and is always positive; therefore, the inequality has no solution, and none of the provided options A, B, C, or D is correct for this part.