Question

1. $45xy - 18x^{2}-7y^{2}$

Explanation:

Step1: Factor out the greatest common factor (GCF)

First, find the GCF of the coefficients 45, -18, and -7? Wait, maybe there's a typo, perhaps the expression is \(45xy - 18x^{2}-7y^{2}\) or maybe \(45x^{2}-18xy - 7y^{2}\)? Let's assume it's a quadratic in two variables, maybe a typo and the first term is \(45x^{2}\) instead of \(45xy\) (since the other terms are quadratic in \(x\) and \(y\)). So let's take the expression as \(45x^{2}-18xy - 7y^{2}\). The GCF of 45, 18, 7? Wait, 45, 18 have GCF 9? No, 45 and 18: GCF is 9? 45=9*5, 18=9*2, but -7 is prime. Wait, maybe it's a trinomial to factor. Let's use the ac method. For \(ax^{2}+bx + c\) (here \(a = 45\), \(b=-18\), \(c=-7\)). So \(ac=45*(-7)=-315\). We need two numbers that multiply to -315 and add to -18. Let's find factors of -315: 15 and -21: 15*(-21)=-315, 15+(-21)=-6. No. 21 and -15: 21*(-15)=-315, 21+(-15)=6. No. 35 and -9: 35*(-9)=-315, 35+(-9)=26. No. 45 and -7: 45*(-7)=-315, 45+(-7)=38. No. Wait, maybe the original expression is \(45x^{2}-18x - 7\)? No, the last term is \(y^{2}\). Wait, maybe the user made a typo. Alternatively, if the expression is \(45xy - 18x^{2}-7y^{2}\), let's rearrange it as \(-18x^{2}+45xy - 7y^{2}\), multiply both sides by -1: \(18x^{2}-45xy + 7y^{2}\). Now, \(a = 18\), \(b=-45\), \(c = 7\). \(ac=18*7 = 126\). Find two numbers that multiply to 126 and add to -45. Factors of 126: -6 and -39? No. -9 and -36? No. -14 and -31? No. Wait, 18x² -45xy +7y². Let's try to factor: (ax - by)(cx - dy) = acx² - (ad + bc)xy + bdy². So ac=18, bd=7, ad + bc=45. Since 7 is prime, b=7, d=1 or b=1, d=7. Let's try b=7, d=1. Then ac=18, ad + bc = a*1 + c*7 = a + 7c = 45. And ac=18. So solve a + 7c = 45 and ac=18. Let's see, c=18/a. So a + 7*(18/a)=45. Multiply by a: a² + 126 = 45a → a² -45a +126=0. Discriminant: 2025 - 504=1521=39². So a=(45±39)/2. a=(45+39)/2=84/2=42, or a=(45-39)/2=6/2=3. If a=3, then c=18/3=6. Check a +7c=3 +42=45. Yes! So (3x -7y)(6x -1y) = 18x² -3xy -42xy +7y²=18x² -45xy +7y². So the original expression (after multiplying by -1) is 18x² -45xy +7y², so the original expression \(-18x^{2}+45xy -7y^{2}= - (18x^{2}-45xy +7y^{2})= - (3x -7y)(6x - y)\). Alternatively, if the original expression is \(45x^{2}-18xy -7y^{2}\), let's try ac method: a=45, c=-7, ac=-315. Find two numbers that multiply to -315 and add to -18. Let's see, 15 and -21: 15*(-21)=-315, 15+(-21)=-6. No. 21 and -15: 6. No. 35 and -9: 26. No. 45 and -7: 38. No. Wait, maybe the first term is \(45x^{2}\), second -18xy, third -7y². Let's try to factor as (ax + by)(cx + dy) = acx² + (ad + bc)xy + bdy². So ac=45, bd=-7, ad + bc=-18. Since -7 is prime, b=7, d=-1 or b=-7, d=1. Let's try b=7, d=-1. Then ac=45, ad + bc = a*(-1) + c*7 = -a +7c = -18. And ac=45. So -a +7c = -18 → a=7c +18. Then ac= (7c +18)c=45 →7c² +18c -45=0. Discriminant: 324 + 1260=1584, which is not a perfect square. Try b=-7, d=1. Then ad + bc = a*1 + c*(-7)=a -7c = -18. ac=45. So a=7c -18. Then (7c -18)c=45 →7c² -18c -45=0. Discriminant: 324 + 1260=1584, not a perfect square. So maybe the original expression has a typo. Assuming the correct expression is \(45x^{2}-18xy -7y^{2}\) is not factorable nicely, but if it's \(45x^{2}-18x -7\) (a quadratic in x), then a=45, b=-18, c=-7. ac=-315. Find two numbers: 15 and -21 (15*(-21)=-315, 15+(-21)=-6). No. 21 and -15: 6. No. 35 and -9: 26. No. 45 and -7: 38. No. Wait, maybe the original problem is to factor \(45x^{2}-18xy -7y^{2}\) and there's a typo, or maybe the first term is \(45xy\) and it's a different problem. Alternatively, maybe the expression is \(45x^{2}-18x^{2}-7y^{2}\), which simplifies to \(27x^{2}-7y^{2}\), a difference of squares? \(27x^{2}=(\sqrt{27}x)^{2}=(3\sqrt{3}x)^{2}\), \(7y^{2}=(\sqrt{7}y)^{2}\), so \(27x^{2}-7y^{2}=(3\sqrt{3}x - \sqrt{7}y)(3\sqrt{3}x + \sqrt{7}y)\). But that's with radicals. Maybe the original expression is \(45x^{2}-18y^{2}-7x^{2}\)? No, the user wrote "45xy -18x² -7y²" (assuming the first term is xy). Let's take the expression as \(45xy - 18x^{2}-7y^{2}\), rearrange to \(-18x^{2}+45xy -7y^{2}\), factor out -1: \(18x^{2}-45xy +7y^{2}\), then factor as (3x -7y)(6x - y) as above, so the original is \(-(3x -7y)(6x - y)=(7y -3x)(6x - y)\) or \(-(3x -7y)(6x - y)\).

Step2: Verify the factoring

Let's expand \(-(3x -7y)(6x - y)\): first expand (3x -7y)(6x - y)=18x² -3xy -42xy +7y²=18x² -45xy +7y², then multiply by -1: -18x² +45xy -7y², which matches the rearranged original expression (if original is 45xy -18x² -7y², which is the same as -18x² +45xy -7y²). So the factored form is \(-(3x -7y)(6x - y)\) or \((7y - 3x)(6x - y)\).

Answer:

\(\boxed{-(3x - 7y)(6x - y)}\) (or equivalent factored form)