Worksheet with ten binomial multiplication problems and their correct solutions.

Let's go through each of these binomial multiplication problems step by step, using the FOIL method (First, Outer, Inner, Last) to multiply two binomials.

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FOIL Method Recap:

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

Then combine like terms.

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Now let’s verify and explain each problem:

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1. $(2x + y)(x - 2y)$

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

Add them:
$2x^2 - 4xy + xy - 2y^2 = 2x^2 - 3xy - 2y^2$ ✔

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2. $(x - 3y)(2x - y)$

- F: $x \cdot 2x = 2x^2$
- O: $x \cdot (-y) = -xy$
- I: $-3y \cdot 2x = -6xy$
- L: $-3y \cdot (-y) = 3y^2$

Add:
$2x^2 - xy - 6xy + 3y^2 = 2x^2 - 7xy + 3y^2$ ✔

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3. $(x - 4)(2x - 1)$

- F: $x \cdot 2x = 2x^2$
- O: $x \cdot (-1) = -x$
- I: $-4 \cdot 2x = -8x$
- L: $-4 \cdot (-1) = 4$

Add:
$2x^2 - x - 8x + 4 = 2x^2 - 9x + 4$ ✔

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

- F: $x \cdot 2x = 2x^2$
- O: $x \cdot 3 = 3x$
- I: $2 \cdot 2x = 4x$
- L: $2 \cdot 3 = 6$

Add:
$2x^2 + 3x + 4x + 6 = 2x^2 + 7x + 6$ ✔

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5. $(2x + 4)(x - 3)$

- F: $2x \cdot x = 2x^2$
- O: $2x \cdot (-3) = -6x$
- I: $4 \cdot x = 4x$
- L: $4 \cdot (-3) = -12$

Add:
$2x^2 - 6x + 4x - 12 = 2x^2 - 2x - 12$ ✔

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6. $(2x + 1)(x + 2)$

- F: $2x \cdot x = 2x^2$
- O: $2x \cdot 2 = 4x$
- I: $1 \cdot x = x$
- L: $1 \cdot 2 = 2$

Add:
$2x^2 + 4x + x + 2 = 2x^2 + 5x + 2$ ✔

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7. $(3x - 3y)(2x - y)$

- F: $3x \cdot 2x = 6x^2$
- O: $3x \cdot (-y) = -3xy$
- I: $-3y \cdot 2x = -6xy$
- L: $-3y \cdot (-y) = 3y^2$

Add:
$6x^2 - 3xy - 6xy + 3y^2 = 6x^2 - 9xy + 3y^2$ ✔

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8. $(x - 3)(2x - 1)$

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

Add:
$2x^2 - x - 6x + 3 = 2x^2 - 7x + 3$

✘ But the answer given is $6x^2 - 9xy + 3y^2$, which is wrong — this is for a different problem.

Wait! There’s a mistake in the image.

Problem 8 is $(x - 3)(2x - 1)$ → should be:
$$
2x^2 - 7x + 3
$$
But the red answer says: $6x^2 - 9xy + 3y^2$ — that's not correct.

This looks like it was copied from Problem 7 or another one. So Problem 8 has an error.

✔ Correct answer: $2x^2 - 7x + 3$

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9. $(x + 2y)(2x - y)$

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

Add:
$2x^2 - xy + 4xy - 2y^2 = 2x^2 + 3xy - 2y^2$ ✔

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10. $(2x + 1)(x + 2)$

This is identical to Problem 6, so same result:
$$
2x^2 + 5x + 2
$$ ✔

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✔ Summary of Corrections:

| Problem | Given Answer | Correct Answer | Notes |
|--------|--------------|----------------|-------|
| 1 | $2x^2 - 3xy - 2y^2$ | ✔ Correct | |
| 2 | $2x^2 - 7xy + 3y^2$ | ✔ Correct | |
| 3 | $2x^2 - 9x + 4$ | ✔ Correct | |
| 4 | $2x^2 + 7x + 6$ | ✔ Correct | |
| 5 | $2x^2 - 2x - 12$ | ✔ Correct | |
| 6 | $2x^2 + 5x + 2$ | ✔ Correct | |
| 7 | $6x^2 - 9xy + 3y^2$ | ✔ Correct | |
| 8 | $6x^2 - 9xy + 3y^2$ ✘ | $2x^2 - 7x + 3$ | Error — likely copied from #7 |
| 9 | $2x^2 + 3xy - 2y^2$ | ✔ Correct | |
| 10 | $2x^2 + 5x + 2$ | ✔ Correct | |

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🔍 Final Answer:

The only incorrect answer is #8.

> Corrected Answer for #8:
> $(x - 3)(2x - 1) = 2x^2 - 7x + 3$

All others are correctly solved.

Let me know if you'd like a visual explanation or practice worksheet!