8xy (x square +6x-16) divide by 4y (x+8) divide the by the method of factorsisation 

To solve the expression \( \frac{8xy(x^2 + 6x - 16)}{4y(x + 8)} \) using factorization, let's break it down step by step.

### 1. Factorize the Numerator

The numerator is \( 8xy(x^2 + 6x - 16) \).

First, we need to factorize the quadratic expression \( x^2 + 6x - 16 \).

#### Factorizing \( x^2 + 6x - 16 \):

We want to find two numbers that multiply to \(-16\) (the constant term) and add to \(6\) (the coefficient of \(x\)).

Let's try splitting the middle term:

1. Multiply the constant term by the coefficient of \(x^2\): \(1 \times -16 = -16\).
2. Find factors of \(-16\) that add up to \(6\): These factors are \(8\) and \(-2\) because \(8 + (-2) = 6\).

So, we can write:

\[ x^2 + 6x - 16 = x^2 + 8x - 2x - 16 \]

Now, group the terms:

\[ (x^2 + 8x) - (2x + 16) \]

Factor out the common terms:

\[ x(x + 8) - 2(x + 8) \]

Now factor out the common binomial factor:

\[ (x - 2)(x + 8) \]

So, the factorization of \( x^2 + 6x - 16 \) is:

\[ x^2 + 6x - 16 = (x - 2)(x + 8) \]

### 2. Substitute the Factorized Form into the Numerator

Now, substitute the factorized quadratic back into the expression:

\[ 8xy(x^2 + 6x - 16) = 8xy(x - 2)(x + 8) \]

### 3. Simplify the Division

Now, let's rewrite the full expression with the factorized numerator:

\[ \frac{8xy(x - 2)(x + 8)}{4y(x + 8)} \]

Next, we can cancel out the common terms:

- \( y \) in the numerator and denominator.
- \( (x + 8) \) in the numerator and denominator.

So, the expression simplifies to:

\[ \frac{8x(x - 2)}{4} \]

### 4. Simplify the Remaining Expression

Finally, simplify by dividing the constants:

\[ \frac{8x(x - 2)}{4} = 2x(x - 2) \]

So, the simplified expression is:

\[ 2x(x - 2) \]

### Final Answer
The expression simplifies to \( \boxed{2x(x - 2)} \).

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**Would you like more details on any step or further assistance?**

### Related Questions:
1. How do you factorize trinomials?
2. What is the process for canceling terms in algebraic fractions?
3. How do you factorize expressions with multiple variables?
4. What are the methods for simplifying rational expressions?
5. How can factorization be applied in solving quadratic equations?
6. What are the rules for simplifying algebraic fractions?
7. How do you solve algebraic expressions involving division?
8. What techniques are useful for factoring complex polynomials?

**Tip:** When dividing expressions, always look for common factors that can be canceled to simplify the problem before performing the division.

Learn how to simplify the expression 8xy(x^2 + 6x - 16) / 4y(x + 8) step-by-step using factorization. This problem covers algebraic concepts such as factorization of polynomials and simplification of rational expressions. Suitable for high school students.

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