$$ x ^ { 2 } + 5 x y - 3 b y $$

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UpStudy AI · Accepted Answer

The expression you've provided is:

$$
x^{2} + 5xy - 3by
$$

Let's analyze and manipulate this expression in a few different ways.

### 1. **Grouping Terms**

You can group the terms to make factoring easier:

$$
x^{2} + 5xy - 3by = x^{2} + (5x - 3b)y
$$

### 2. **Factoring by Common Terms**

Looking for common factors in the terms:

- The first two terms, $$x^{2}$$ and $$5xy$$, share a common factor of $$x$$.
- The last term, $$-3by$$, shares a factor of $$y$$ with the second term.

Factoring $$x$$ from the first two terms:

$$
x(x + 5y) - 3by
$$

Now, you can see if further factoring is possible. Notice that both remaining terms contain a $$y$$:

$$
x(x + 5y) - 3b y = y( \, x \cdot \frac{(x + 5y)}{y} \, ) - 3b y
$$

However, since $$x(x + 5y)$$ isn't easily factorizable with $$-3by$$, the expression in its factored form is:

$$
x(x + 5y) - 3by
$$

### 3. **Rearranging Terms**

Sometimes rearranging terms can make patterns more evident:

$$
x^{2} + 5xy - 3by = x^{2} + 5xy - 3by
$$

There's no immediate simplification from rearranging in this case.

### 4. **Completing the Square**

If you're interested in completing the square for the quadratic in $$x$$:

Consider the expression as a quadratic in $$x$$:

$$
x^{2} + 5xy - 3by = x^{2} + 5y \cdot x - 3b y
$$

To complete the square:

1. **Identify coefficients:**
   - $$a = 1$$
   - $$b = 5y$$
   - $$c = -3b y$$

2. **Compute $$\left(\frac{b}{2}\right)^2 = \left(\frac{5y}{2}\right)^2 = \frac{25y^2}{4}$$

3. **Add and subtract this inside the expression:**

$$
x^{2} + 5xy + \frac{25y^2}{4} - \frac{25y^2}{4} - 3by = \left(x + \frac{5y}{2}\right)^2 - \frac{25y^2}{4} - 3by
$$

4. **Combine like terms:**

$$
\left(x + \frac{5y}{2}\right)^2 - \left(\frac{25y^2}{4} + 3b y\right)
$$

This form is useful for solving equations or analyzing the expression's properties.

### 5. **Possible Applications**

Depending on the context, this expression could represent:

- **A quadratic equation** in terms of $$x$$ or $$y$$.
- **A term in a larger algebraic expression** that might require further manipulation.
- **Part of a function** where you might be asked to find maxima, minima, or roots.

### 6. **Next Steps**

If you have a specific question or a particular operation you'd like to perform on this expression—such as solving for one variable in terms of others, graphing, or applying it to a real-world scenario—please provide more details, and I'd be happy to help!
The expression can be factored as $$ x(x + 5y) - 3by $$.

\( x ^ { 2 } + 5 x y - 3 b y \)

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