Complete Guide to Multiplying Binomials: Foil Method and Box ... - Free Printable

It looks like you've uploaded a worksheet titled "Practice: Multiplying Binomials", with 20 problems asking to multiply two binomials and write the result in expanded form:
$$
ax^2 + bx + c
$$

Additionally, there's a small image in the bottom right showing an example:
- $ x = -5 $
- $ 6x^2 - 30x + 12 $
- With values: $ 6x^2 = 150 $, $ -30x = 150 $, $ +12 = 60 $, leading to $ 150 + 150 + 60 = 360 $

But since I can't view images directly, I’ll solve all 20 problems step-by-step using the FOIL method (First, Outer, Inner, Last), and then combine like terms to get the final answer in standard form.

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🔷 FOIL Method Reminder:

For $(a + b)(c + d)$:
- F: First → $a \cdot c$
- O: Outer → $a \cdot d$
- I: Inner → $b \cdot c$
- L: Last → $b \cdot d$

Then add all four parts and simplify.

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Let’s go through each problem:

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1) $(5x - 2)(2x + 5)$

- F: $5x \cdot 2x = 10x^2$
- O: $5x \cdot 5 = 25x$
- I: $-2 \cdot 2x = -4x$
- L: $-2 \cdot 5 = -10$
- Combine: $10x^2 + (25x - 4x) - 10 = 10x^2 + 21x - 10$

✔ Answer: $10x^2 + 21x - 10$

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2) $(2x - 3)(x + 4)$

- F: $2x \cdot x = 2x^2$
- O: $2x \cdot 4 = 8x$
- I: $-3 \cdot x = -3x$
- L: $-3 \cdot 4 = -12$
- Combine: $2x^2 + (8x - 3x) - 12 = 2x^2 + 5x - 12$

✔ Answer: $2x^2 + 5x - 12$

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3) $(2m + 1)(5m + 5)$

- F: $2m \cdot 5m = 10m^2$
- O: $2m \cdot 5 = 10m$
- I: $1 \cdot 5m = 5m$
- L: $1 \cdot 5 = 5$
- Combine: $10m^2 + (10m + 5m) + 5 = 10m^2 + 15m + 5$

✔ Answer: $10m^2 + 15m + 5$

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4) $(3u + 3)(5u - 1)$

- F: $3u \cdot 5u = 15u^2$
- O: $3u \cdot (-1) = -3u$
- I: $3 \cdot 5u = 15u$
- L: $3 \cdot (-1) = -3$
- Combine: $15u^2 + (-3u + 15u) - 3 = 15u^2 + 12u - 3$

✔ Answer: $15u^2 + 12u - 3$

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5) $(4k + 3)(5k + 1)$

- F: $4k \cdot 5k = 20k^2$
- O: $4k \cdot 1 = 4k$
- I: $3 \cdot 5k = 15k$
- L: $3 \cdot 1 = 3$
- Combine: $20k^2 + (4k + 15k) + 3 = 20k^2 + 19k + 3$

✔ Answer: $20k^2 + 19k + 3$

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6) $(3r + 3)(5r + 1)$

- F: $3r \cdot 5r = 15r^2$
- O: $3r \cdot 1 = 3r$
- I: $3 \cdot 5r = 15r$
- L: $3 \cdot 1 = 3$
- Combine: $15r^2 + (3r + 15r) + 3 = 15r^2 + 18r + 3$

✔ Answer: $15r^2 + 18r + 3$

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7) $(5x - 4)(5x + 3)$

- F: $5x \cdot 5x = 25x^2$
- O: $5x \cdot 3 = 15x$
- I: $-4 \cdot 5x = -20x$
- L: $-4 \cdot 3 = -12$
- Combine: $25x^2 + (15x - 20x) - 12 = 25x^2 - 5x - 12$

✔ Answer: $25x^2 - 5x - 12$

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8) $(3r + 3)(2r - 1)$

- F: $3r \cdot 2r = 6r^2$
- O: $3r \cdot (-1) = -3r$
- I: $3 \cdot 2r = 6r$
- L: $3 \cdot (-1) = -3$
- Combine: $6r^2 + (-3r + 6r) - 3 = 6r^2 + 3r - 3$

✔ Answer: $6r^2 + 3r - 3$

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9) $(3m - 3)(3m + 5)$

- F: $3m \cdot 3m = 9m^2$
- O: $3m \cdot 5 = 15m$
- I: $-3 \cdot 3m = -9m$
- L: $-3 \cdot 5 = -15$
- Combine: $9m^2 + (15m - 9m) - 15 = 9m^2 + 6m - 15$

✔ Answer: $9m^2 + 6m - 15$

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10) $(4x + 2)(x - 1)$

- F: $4x \cdot x = 4x^2$
- O: $4x \cdot (-1) = -4x$
- I: $2 \cdot x = 2x$
- L: $2 \cdot (-1) = -2$
- Combine: $4x^2 + (-4x + 2x) - 2 = 4x^2 - 2x - 2$

✔ Answer: $4x^2 - 2x - 2$

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11) $(-2n - 6)(5n - 3)$

- F: $-2n \cdot 5n = -10n^2$
- O: $-2n \cdot (-3) = 6n$
- I: $-6 \cdot 5n = -30n$
- L: $-6 \cdot (-3) = 18$
- Combine: $-10n^2 + (6n - 30n) + 18 = -10n^2 - 24n + 18$

✔ Answer: $-10n^2 - 24n + 18$

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12) $(-7h + 7)(-6h - 6)$

- F: $-7h \cdot -6h = 42h^2$
- O: $-7h \cdot (-6) = 42h$
- I: $7 \cdot (-6h) = -42h$
- L: $7 \cdot (-6) = -42$
- Combine: $42h^2 + (42h - 42h) - 42 = 42h^2 + 0h - 42 = 42h^2 - 42$

✔ Answer: $42h^2 - 42$

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13) $(4b - 8)(-8b - 8)$

- F: $4b \cdot (-8b) = -32b^2$
- O: $4b \cdot (-8) = -32b$
- I: $-8 \cdot (-8b) = 64b$
- L: $-8 \cdot (-8) = 64$
- Combine: $-32b^2 + (-32b + 64b) + 64 = -32b^2 + 32b + 64$

✔ Answer: $-32b^2 + 32b + 64$

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14) $(7a - 2)(-2a + 7)$

- F: $7a \cdot (-2a) = -14a^2$
- O: $7a \cdot 7 = 49a$
- I: $-2 \cdot (-2a) = 4a$
- L: $-2 \cdot 7 = -14$
- Combine: $-14a^2 + (49a + 4a) - 14 = -14a^2 + 53a - 14$

✔ Answer: $-14a^2 + 53a - 14$

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15) $(2k + 5)(-8k + 5)$

- F: $2k \cdot (-8k) = -16k^2$
- O: $2k \cdot 5 = 10k$
- I: $5 \cdot (-8k) = -40k$
- L: $5 \cdot 5 = 25$
- Combine: $-16k^2 + (10k - 40k) + 25 = -16k^2 - 30k + 25$

✔ Answer: $-16k^2 - 30k + 25$

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16) $(-a - 2)(-7a + 7)$

- F: $-a \cdot (-7a) = 7a^2$
- O: $-a \cdot 7 = -7a$
- I: $-2 \cdot (-7a) = 14a$
- L: $-2 \cdot 7 = -14$
- Combine: $7a^2 + (-7a + 14a) - 14 = 7a^2 + 7a - 14$

✔ Answer: $7a^2 + 7a - 14$

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17) $(-8n - 4)(-4n + 3)$

- F: $-8n \cdot (-4n) = 32n^2$
- O: $-8n \cdot 3 = -24n$
- I: $-4 \cdot (-4n) = 16n$
- L: $-4 \cdot 3 = -12$
- Combine: $32n^2 + (-24n + 16n) - 12 = 32n^2 - 8n - 12$

✔ Answer: $32n^2 - 8n - 12$

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18) $(-4n - 6)(4n - 8)$

- F: $-4n \cdot 4n = -16n^2$
- O: $-4n \cdot (-8) = 32n$
- I: $-6 \cdot 4n = -24n$
- L: $-6 \cdot (-8) = 48$
- Combine: $-16n^2 + (32n - 24n) + 48 = -16n^2 + 8n + 48$

✔ Answer: $-16n^2 + 8n + 48$

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19) $(5m - 1)(7m + 8)$

- F: $5m \cdot 7m = 35m^2$
- O: $5m \cdot 8 = 40m$
- I: $-1 \cdot 7m = -7m$
- L: $-1 \cdot 8 = -8$
- Combine: $35m^2 + (40m - 7m) - 8 = 35m^2 + 33m - 8$

✔ Answer: $35m^2 + 33m - 8$

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20) $(7n - 2)(-n + 8)$

- F: $7n \cdot (-n) = -7n^2$
- O: $7n \cdot 8 = 56n$
- I: $-2 \cdot (-n) = 2n$
- L: $-2 \cdot 8 = -16$
- Combine: $-7n^2 + (56n + 2n) - 16 = -7n^2 + 58n - 16$

✔ Answer: $-7n^2 + 58n - 16$

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $10x^2 + 21x - 10$ |
| 2 | $2x^2 + 5x - 12$ |
| 3 | $10m^2 + 15m + 5$ |
| 4 | $15u^2 + 12u - 3$ |
| 5 | $20k^2 + 19k + 3$ |
| 6 | $15r^2 + 18r + 3$ |
| 7 | $25x^2 - 5x - 12$ |
| 8 | $6r^2 + 3r - 3$ |
| 9 | $9m^2 + 6m - 15$ |
| 10 | $4x^2 - 2x - 2$ |
| 11 | $-10n^2 - 24n + 18$ |
| 12 | $42h^2 - 42$ |
| 13 | $-32b^2 + 32b + 64$ |
| 14 | $-14a^2 + 53a - 14$ |
| 15 | $-16k^2 - 30k + 25$ |
| 16 | $7a^2 + 7a - 14$ |
| 17 | $32n^2 - 8n - 12$ |
| 18 | $-16n^2 + 8n + 48$ |
| 19 | $35m^2 + 33m - 8$ |
| 20 | $-7n^2 + 58n - 16$ |

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📌 Bonus: About the Example in Bottom Right

You showed:
- $ x = -5 $
- Expression: $6x^2 - 30x + 12$
- Then calculated:
- $6x^2 = 6 \cdot 25 = 150$
- $-30x = -30 \cdot (-5) = 150$
- $+12 = 12$
- Total: $150 + 150 + 12 = 312$

Wait — your image says 60, but that seems incorrect. Let me check:

If $x = -5$:
- $6x^2 = 6 \cdot 25 = 150$
- $-30x = -30 \cdot (-5) = +150$
- $+12 = 12$
- Sum: $150 + 150 + 12 = \boxed{312}$

So either:
- The expression is not $6x^2 - 30x + 12$, or
- The value of $x$ is different, or
- There was a typo in the image.

But based on what’s shown, it should be 312, not 60.

Let me know if you want help checking that part further!

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Let me know if you'd like these answers formatted for printing or need explanations for any specific problem!