Combine like terms to simplify the expression: (2x + 3y - 4x + 5y = -2x + 8y).

Add 5 to both sides of the equation: (3x - 5 + 5 = 10 + 5). Simplify: (3x = 15). Divide both sides by 3: (x = 15/3). Simplify: (x = 5).

Find two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3. Rewrite the polynomial as (x^2 + 2x + 3x + 6). Group the terms: ((x^2 + 2x) + (3x + 6)). Factor each group: (x(x + 2) + 3(x + 2)). Combine the factors: ((x + 2)(x + 3)).

Add 3 to both sides of the inequality: (2x - 3 + 3 < 7 + 3). Simplify: (2x < 10). Divide both sides by 2: (x < 10/2). Simplify: (x < 5).

Simplify each square root: (\sqrt{16} = 4) and (\sqrt{25} = 5). Add the simplified values: (4 + 5 = 10).

Distribute the 2: (2x + 6 = 10). Subtract 6 from both sides: (2x + 6 - 6 = 10 - 6). Simplify: (2x = 4). Divide both sides by 2: (2x/2 = 4/2). Simplify: (x = 2).

This is a difference of squares. Rewrite the polynomial as (x^2 - 3^2). Use the formula (a^2 - b^2 = (a + b)(a - b)). Substitute (a = x) and (b = 3): (x^2 - 3^2 = (x + 3)(x - 3)).

Subtract 4 from both sides of the inequality: (3x + 4 - 4 > 16 - 4). Simplify: (3x > 12). Divide both sides by 3: (3x/3 > 12/3). Simplify: (x > 4).

Combine like terms: (\frac{1}{2}x + \frac{1}{4}x - \frac{1}{3}y - \frac{1}{6}y = \frac{3}{4}x - \frac{1}{2}y).

Distribute the 4: (8x - 4 = 20). Add 4 to both sides: (8x - 4 + 4 = 20 + 4). Simplify: (8x = 24). Divide both sides by 8: (8x/8 = 24/8). Simplify: (x = 3).

Find two numbers that add up to 7 and multiply to 12. These numbers are 3 and 4. Rewrite the polynomial as (x^2 + 3x + 4x + 12). Group the terms: ((x^2 + 3x) + (4x + 12)). Factor each group: (x(x + 3) + 4(x + 3)). Combine the factors: ((x + 3)(x + 4)).

Subtract 2x from both sides of the inequality: (2x - 5 - 2x < 3x + 2 - 2x). Simplify: (-5 < x + 2). Subtract 2 from both sides: (-5 - 2 < x + 2 - 2). Simplify: (-7 < x).

Combine like terms: (\frac{2}{3}x - \frac{1}{6}x + \frac{1}{2}y + \frac{3}{4}y = \frac{5}{6}x + \frac{7}{6}y).

Distribute the 3: (3x - 6 = 15). Add 6 to both sides: (3x - 6 + 6 = 15 + 6). Simplify: (3x = 21). Divide both sides by 3: (3x/3 = 21/3). Simplify: (x = 7).

Find two numbers that add up to -4 and multiply to -21. These numbers are -7 and 3. Rewrite the polynomial as (x^2 - 7x + 3x - 21). Group the terms: ((x^2 - 7x) + (3x - 21)). Factor each group: (x(x - 7) + 3(x - 7)). Combine the factors: ((x + 3)(x - 7)).