(1) \( x^{2}+3 x-18 \) (J) \( x^{2}-17 x-18 \) (K) \( x^{2}+5 x-24 \) (L) \( x^{2}-10 x-24 \) (M) \( x^{2}+2 x y-15 y^{2} \) (N) \( x^{2}-5 x y-50 y^{2} \) (O) \( x^{2}-9 x y-36 y^{2} \) (P) \( x^{2}+5 x y-36 y^{2} \)

Here’s one way to factor each expression: (1) Factor x² + 3x – 18   We need two numbers that multiply to –18 and add to 3. Since 6 × (–3) = –18 and 6 + (–3) = 3, we have:   x² + 3x – 18 = (x + 6)(x – 3). (J) Factor x² – 17x – 18   Look for two numbers that multiply to –18 and add to –17. The numbers –18 and 1 work because (–18) × 1 = –18 and (–18) + 1 = –17, so:   x² – 17x – 18 = (x – 18)(x + 1). (K) Factor x² + 5x – 24   Find two numbers multiplying to –24 and summing to 5. The pair 8 and –3 works (8 × –3 = –24 and 8 + (–3) = 5):   x² + 5x – 24 = (x + 8)(x – 3). (L) Factor x² – 10x – 24   We need numbers that multiply to –24 and add to –10. The numbers –12 and 2 satisfy this (–12 × 2 = –24 and –12 + 2 = –10):   x² – 10x – 24 = (x – 12)(x + 2). (M) Factor x² + 2xy – 15y²   Treat this as a quadratic in x with y as a constant. We want numbers a and b such that:    a + b = 2 and a·b = –15.   Since 5 + (–3) = 2 and 5 × (–3) = –15, we have:   x² + 2xy – 15y² = (x + 5y)(x – 3y). (N) Factor x² – 5xy – 50y²   Here, find two numbers that satisfy:    Sum = –5 and Product = –50.   The numbers 5 and –10 work because 5 + (–10) = –5 and 5 × (–10) = –50:   x² – 5xy – 50y² = (x + 5y)(x – 10y). (O) Factor x² – 9xy – 36y²   Find two numbers that add to –9 and multiply to –36. The numbers 3 and –12 work (3 + (–12) = –9 and 3 × (–12) = –36):   x² – 9xy – 36y² = (x + 3y)(x – 12y). (P) Factor x² + 5xy – 36y²   We need numbers that add to 5 and multiply to –36. The numbers 9 and –4 meet these conditions (9 + (–4) = 5 and 9 × (–4) = –36):   x² + 5xy – 36y² = (x + 9y)(x – 4y). Each answer gives a correct factorization of the corresponding quadratic expression.