Simplify binomials using the FOIL method with this algebra worksheet from Diamond Tutoring.

To solve the given binomials using the FOIL method, we will follow these steps:

1. First: Multiply the first terms in each binomial.
2. Outside: Multiply the outer terms in the product.
3. Inside: Multiply the inner terms in the product.
4. Last: Multiply the last terms in each binomial.
5. Combine like terms to simplify the expression.

Let's solve each problem step by step.

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1. \((x + 2)(2x + 5)\)

- F: \(x \cdot 2x = 2x^2\)
- O: \(x \cdot 5 = 5x\)
- I: \(2 \cdot 2x = 4x\)
- L: \(2 \cdot 5 = 10\)

Combine all terms:
\[
2x^2 + 5x + 4x + 10 = 2x^2 + 9x + 10
\]

Answer: \(\boxed{2x^2 + 9x + 10}\)

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2. \((3x - 3)(x + 3)\)

- F: \(3x \cdot x = 3x^2\)
- O: \(3x \cdot 3 = 9x\)
- I: \(-3 \cdot x = -3x\)
- L: \(-3 \cdot 3 = -9\)

Combine all terms:
\[
3x^2 + 9x - 3x - 9 = 3x^2 + 6x - 9
\]

Answer: \(\boxed{3x^2 + 6x - 9}\)

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3. \((4a - 5)(a + 3)\)

- F: \(4a \cdot a = 4a^2\)
- O: \(4a \cdot 3 = 12a\)
- I: \(-5 \cdot a = -5a\)
- L: \(-5 \cdot 3 = -15\)

Combine all terms:
\[
4a^2 + 12a - 5a - 15 = 4a^2 + 7a - 15
\]

Answer: \(\boxed{4a^2 + 7a - 15}\)

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4. \((2a + 1)(a + 3)\)

- F: \(2a \cdot a = 2a^2\)
- O: \(2a \cdot 3 = 6a\)
- I: \(1 \cdot a = a\)
- L: \(1 \cdot 3 = 3\)

Combine all terms:
\[
2a^2 + 6a + a + 3 = 2a^2 + 7a + 3
\]

Answer: \(\boxed{2a^2 + 7a + 3}\)

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5. \((2x + 7)(x - 8)\)

- F: \(2x \cdot x = 2x^2\)
- O: \(2x \cdot (-8) = -16x\)
- I: \(7 \cdot x = 7x\)
- L: \(7 \cdot (-8) = -56\)

Combine all terms:
\[
2x^2 - 16x + 7x - 56 = 2x^2 - 9x - 56
\]

Answer: \(\boxed{2x^2 - 9x - 56}\)

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6. \((x + 2)^2\)

This is the same as \((x + 2)(x + 2)\).

- F: \(x \cdot x = x^2\)
- O: \(x \cdot 2 = 2x\)
- I: \(2 \cdot x = 2x\)
- L: \(2 \cdot 2 = 4\)

Combine all terms:
\[
x^2 + 2x + 2x + 4 = x^2 + 4x + 4
\]

Answer: \(\boxed{x^2 + 4x + 4}\)

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7. \((3x + 1)(2x + 6)\)

- F: \(3x \cdot 2x = 6x^2\)
- O: \(3x \cdot 6 = 18x\)
- I: \(1 \cdot 2x = 2x\)
- L: \(1 \cdot 6 = 6\)

Combine all terms:
\[
6x^2 + 18x + 2x + 6 = 6x^2 + 20x + 6
\]

Answer: \(\boxed{6x^2 + 20x + 6}\)

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8. \((a - 5)^2\)

This is the same as \((a - 5)(a - 5)\).

- F: \(a \cdot a = a^2\)
- O: \(a \cdot (-5) = -5a\)
- I: \(-5 \cdot a = -5a\)
- L: \(-5 \cdot (-5) = 25\)

Combine all terms:
\[
a^2 - 5a - 5a + 25 = a^2 - 10a + 25
\]

Answer: \(\boxed{a^2 - 10a + 25}\)

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Final Answers:

1. \(\boxed{2x^2 + 9x + 10}\)
2. \(\boxed{3x^2 + 6x - 9}\)
3. \(\boxed{4a^2 + 7a - 15}\)
4. \(\boxed{2a^2 + 7a + 3}\)
5. \(\boxed{2x^2 - 9x - 56}\)
6. \(\boxed{x^2 + 4x + 4}\)
7. \(\boxed{6x^2 + 20x + 6}\)
8. \(\boxed{a^2 - 10a + 25}\)