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### Factorization of Algebraic Expressions

Factorize the following expressions:

[tex]\[
\begin{array}{l}
1. \ x^2 + 6x + 5 - 4y - y^2 \\
2. \ p^2 - 12p - 28 + 16q - q^2 \\
3. \ 9a^2 - 30a + 24 - 8x - 16x^2 \\
4. \ 16p^2 - 72pq + 80q^2 - 6qr - 9r^2 \\
\end{array}
\][/tex]

Additional expressions to factorize:

[tex]\[
\begin{array}{l}
5. \ a^2 - 10a + 16 - 6b - b^2 \\
6. \ x^4 + 8x^2 - 65 + 18y - y^2 \\
7. \ 625y^2 + 400y - 36 + 20z - z^2 \\
8. \ 25x^2 - 20xy - 21y^2 + 10yz - z^2 \\
\end{array}
\][/tex]

Solve the following into factors:

[tex]\[
\begin{array}{l}
1. \ (a^2 - b^2)(c^2 - d^2) + 4abcd \\
2. \ (p^2 - 4)(9 - q^2) + 24pq \\
3. \ (x^2 - 1)(y^2 - 1) - 4xy \\
4. \ (9 - x^2)(100 - y^2) - 120xy \\
\end{array}
\][/tex]

### Geometry Problems

1. A square sheet of paper is [tex]\(25 \text{ cm}\)[/tex] long. A small square portion of length [tex]\(9 \text{ cm}\)[/tex] is cut out from it. Find the area of the remaining portion of the paper.

2. A farmer has a square field of length [tex]\(150 \text{ m}\)[/tex]. He separates a small square portion of length [tex]\(60 \text{ m}\)[/tex] from it to cultivate vegetables, and he cultivates crops in the remaining portion. Find the area of the crops-cultivated portion of the field.



Answer :

GinnyAnswer
Let's begin by solving the problem step-by-step:

Given:
- A square sheet of paper with side length of [tex]\(25 \text{ cm}\)[/tex].
- A small square portion with side length of [tex]\(9 \text{ cm}\)[/tex] is cut out from it.

To Find:
- The area of the remaining portion of the paper after the small square has been cut out.

1. Calculate the Area of the Original Square Sheet:

The area [tex]\(A\)[/tex] of a square is given by the formula:
[tex]\[ A = \text{side length}^2 \][/tex]
For the original square sheet,

[tex]\[ A_{\text{original}} = 25 \text{ cm} \times 25 \text{ cm} = 625 \text{ cm}^2 \][/tex]

2. Calculate the Area of the Small Square that is Cut Out:

Similarly, for the small square,

[tex]\[ A_{\text{cut-out}} = 9 \text{ cm} \times 9 \text{ cm} = 81 \text{ cm}^2 \][/tex]

3. Calculate the Area of the Remaining Portion of the Paper:

The area of the remaining portion is the area of the original square sheet minus the area of the small square cut out:

[tex]\[ A_{\text{remaining}} = A_{\text{original}} - A_{\text{cut-out}} \][/tex]
Substitute the respective areas calculated:

[tex]\[ A_{\text{remaining}} = 625 \text{ cm}^2 - 81 \text{ cm}^2 = 544 \text{ cm}^2 \][/tex]

Therefore, the area of the remaining portion of the paper is [tex]\(544 \text{ cm}^2\)[/tex].

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