To simplify the given expression:

$\frac{3yx^2 + xy^2 - 3y - y^2}{2}$

into the form $(x - 1)(3x + y + 3)y$, we need to work through it step-by-step:

Step 1: Break Down the Expression

The numerator is:

$3yx^2 + xy^2 - 3y - y^2$

We want to see if we can factor this expression in some way.

Step 2: Group Terms

Group the terms in pairs to look for common factors:

$(3yx^2 + xy^2) - (3y + y^2)$

Step 3: Factor Each Group

1. First group: $3yx^2 + xy^2$

   • Factor out $y$: $y(3x^2 + xy)$
   • Factor further: $y \cdot x(3x + y)$

2. Second group: $- (3y + y^2)$

   • Factor out $-y$: $-y(3 + y)$

Step 4: Rewrite the Numerator

Now, we rewrite the numerator by combining these factors:

$y \cdot x(3x + y) - y(3 + y)$

Factor out $y$ from the entire expression:

$y[x(3x + y) - (3 + y)]$

Step 5: Simplify the Expression Inside the Bracket

Now, simplify $x(3x + y) - (3 + y)$:

$x(3x + y) - 3 - y = 3x^2 + xy - 3 - y$

Step 6: Look for the Desired Form

We want this expression to eventually resemble $(x - 1)(3x + y + 3)$. Notice that we are close; we need to find a way to express the terms in that factorized form.

Step 7: Find the Factorization

After trying to simplify and factor, we find that:

$y(3x^2 + xy - 3 - y) = (x - 1)(3x + y + 3)y$

Thus, the expression simplifies to:

$\frac{3yx^2 + xy^2 - 3y - y^2}{2} = (x - 1)(3x + y + 3)y$

Final Simplified Result

The final simplified result is:

$(x - 1)(3x + y + 3)y$

Do you want further details or have any questions?

5 Related Questions:

1. How do you identify common factors in a polynomial?
2. What are the different methods for factoring complex polynomials?
3. How can you verify if two expressions are equivalent?
4. Why is grouping terms useful in polynomial factorization?
5. Can every polynomial be factored in this way?

Tip

When simplifying complex expressions, always look for common factors and try grouping terms strategically to make factoring easier.