To factorize the expression [tex]\( x^2 + 6x + 5 - 4y - y^2 \)[/tex], we need to recognize it in a form where we can apply factoring techniques. Here's how you can do it step by step:
1. Rearrange the Given Expression: Group the terms involving [tex]\( x \)[/tex] and [tex]\( y \)[/tex] together.
[tex]\[
x^2 + 6x + 5 - y^2 - 4y
\][/tex]
2. Reorder the Expression for Clarity: Write it as:
[tex]\[
x^2 + 6x + 5 - y^2 - 4y
\][/tex]
3. Identify and Factor Quadratic Terms: We can see there are two quadratic expressions here, one in [tex]\( x \)[/tex] and one in [tex]\( y \)[/tex]:
[tex]\[
(x^2 + 6x + 5) - (y^2 + 4y)
\][/tex]
4. Complete the Square for Each Quadratic:
- For the quadratic in [tex]\( x \)[/tex]:
[tex]\[
x^2 + 6x + 5 = (x + 3)^2 - 4
\][/tex]
However, directly factoring it, we get:
[tex]\[
x^2 + 6x + 5 = (x + 1)(x + 5)
\][/tex]
- For the quadratic in [tex]\( y \)[/tex], to see the usual factor form:
[tex]\[
y^2 + 4y = (y + 2)^2 - 4
\][/tex]
But typically, we know:
[tex]\[
- y^2 - 4y = - (y^2 + 4y + 4 - 4);
\][/tex]
Combining, we get:
[tex]\[
- (y + 2 - y); usual factor forms: -(y + 1)(y + 4)
\][/tex]
5. Combine Both Expressions via Factoring Sum of two common Statements:
But another Elegant Way:
- Recognize if we can merge overall:
[tex]\[
Simplify and rearrange dynamically the overall addition Common factors like;
\;Comparing factor cross form to together start expressions ( x - something -twe y )
Directly merger.
Putting these together gives us:
The factored form of the given expression is:
\[
\boxed{(x - y + 1)(x + y + 5)}
\][/tex]
