Factor the polynomial x ^ 2-y ^ 2 + 4x + 42.Solve the inequality: 6x² ≤9x

The task consists of two parts with different performances. Let’s do each part separately.

Let’s rewrite this expression as x² + 2 * 2 * x + 2² – y² and denote it by A. Applying to the first three terms the formula of abbreviated multiplication (a + b) ² = a² + 2 * a * b + b² (the square of the sum) , we get A = (x + 2) ² – y². The formula for abbreviated multiplication (a – b) * (a + b) = a² – b² (difference of squares) will allow factoring the last difference into factors: A = (x + 2 – y) * (x + 2 + y). Answer: (x + 2 – y) * (x + 2 + y).
Consider the inequality 6 * х² ≤ 9 * х, which we rewrite as: 6 * х² – 9 * х ≤ 0 or 6 * х * (х – 1.5) ≤ 0, whence х * (х – 1.5) ≤ 0. Applying the method of intervals, we can easily find a solution to the last inequality. It has the form 0 ≤ x ≤ 1.5. Answer: 0 ≤ x ≤ 1.5.