Question

1. $x^{2}-10xy - 11y^{2}$

Explanation:

Step1: Analyze the quadratic in two variables

We have the expression \(x^{2}-10xy - 11y^{2}\). We need to factor it. Let's assume it can be factored as \((x + ay)(x+by)\), where \(a\) and \(b\) are constants. Expanding \((x + ay)(x + by)=x^{2}+(a + b)xy+aby^{2}\). Comparing with \(x^{2}-10xy-11y^{2}\), we have \(a + b=- 10\) and \(ab=-11\).

Step2: Find a and b

We need two numbers that multiply to \(- 11\) and add up to \(-10\). The numbers are \(-11\) and \(1\) since \(-11\times1=-11\) and \(-11 + 1=-10\).

Step3: Factor the expression

So, \(x^{2}-10xy-11y^{2}=(x + y)(x-11y)\)

Answer:

\((x + y)(x - 11y)\)