c) $$ x^{2}+t x y-4 x y-4 t y^{2} $$

Question

UpStudy AI · Accepted Answer

To factor the expression:

$$ x^{2} + txy - 4xy - 4ty^{2} $$

**Step 1: Combine Like Terms**

First, group the terms involving $$ xy $$:

$$ x^{2} + (txy - 4xy) - 4ty^{2} = x^{2} + xy(t - 4) - 4ty^{2} $$

**Step 2: Factor by Grouping**

Treat the expression as a quadratic in terms of $$ x $$:

$$ x^{2} + xy(t - 4) - 4ty^{2} $$

We aim to factor this into the form:

$$ (x + ay)(x + by) $$

where $$ a $$ and $$ b $$ are constants to be determined.

**Step 3: Find $$ a $$ and $$ b $$**

To find $$ a $$ and $$ b $$, set up the following equations based on the expanded form:

1. $$ a + b = t - 4 $$ (sum of the coefficients)
2. $$ ab = -4t $$ (product of the coefficients)

Solving these, we find:

- $$ a = t $$
- $$ b = -4 $$

**Step 4: Write the Factored Form**

Substitute $$ a $$ and $$ b $$ back into the factored form:

$$ (x + ty)(x - 4y) $$

**Final Answer:**

$$ x^{2} + txy - 4xy - 4ty^{2} = (x + ty)(x - 4y) $$
$$ x^{2} + txy - 4xy - 4ty^{2} = (x + ty)(x - 4y) $$

c) \( x^{2}+t x y-4 x y-4 t y^{2} \)