Foil Method Math Problems - Free Printable

Problem Overview:

The task involves using the FOIL method to expand each of the given binomial expressions. The FOIL method stands for:
- F: First terms
- O: Outer terms
- I: Inner terms
- L: Last terms

We will apply this method to each expression systematically.

---

Solution:

#### Step 1: Understand the FOIL Method
For any two binomials \((a + b)(c + d)\):
1. First: Multiply the first terms (\(a \cdot c\)).
2. Outer: Multiply the outer terms (\(a \cdot d\)).
3. Inner: Multiply the inner terms (\(b \cdot c\)).
4. Last: Multiply the last terms (\(b \cdot d\)).

Finally, combine all these products and simplify by combining like terms.

#### Step 2: Solve Each Expression

##### Expression (1): \((x + 3)(x + 4)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 4 = 4x \\
&\text{Inner: } 3 \cdot x = 3x \\
&\text{Last: } 3 \cdot 4 = 12 \\
&\text{Combine: } x^2 + 4x + 3x + 12 = x^2 + 7x + 12
\end{aligned}
\]
Answer: \(x^2 + 7x + 12\)

##### Expression (2): \((x + 5)(x + 2)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 2 = 2x \\
&\text{Inner: } 5 \cdot x = 5x \\
&\text{Last: } 5 \cdot 2 = 10 \\
&\text{Combine: } x^2 + 2x + 5x + 10 = x^2 + 7x + 10
\end{aligned}
\]
Answer: \(x^2 + 7x + 10\)

##### Expression (3): \((x + 7)(x + 6)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 6 = 6x \\
&\text{Inner: } 7 \cdot x = 7x \\
&\text{Last: } 7 \cdot 6 = 42 \\
&\text{Combine: } x^2 + 6x + 7x + 42 = x^2 + 13x + 42
\end{aligned}
\]
Answer: \(x^2 + 13x + 42\)

##### Expression (4): \((x + 5)(x + 6)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 6 = 6x \\
&\text{Inner: } 5 \cdot x = 5x \\
&\text{Last: } 5 \cdot 6 = 30 \\
&\text{Combine: } x^2 + 6x + 5x + 30 = x^2 + 11x + 30
\end{aligned}
\]
Answer: \(x^2 + 11x + 30\)

##### Expression (5): \((x + 8)(x + 4)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 4 = 4x \\
&\text{Inner: } 8 \cdot x = 8x \\
&\text{Last: } 8 \cdot 4 = 32 \\
&\text{Combine: } x^2 + 4x + 8x + 32 = x^2 + 12x + 32
\end{aligned}
\]
Answer: \(x^2 + 12x + 32\)

##### Expression (6): \((x + 10)(x + 3)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 3 = 3x \\
&\text{Inner: } 10 \cdot x = 10x \\
&\text{Last: } 10 \cdot 3 = 30 \\
&\text{Combine: } x^2 + 3x + 10x + 30 = x^2 + 13x + 30
\end{aligned}
\]
Answer: \(x^2 + 13x + 30\)

##### Expression (7): \((x + 9)(x + 6)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 6 = 6x \\
&\text{Inner: } 9 \cdot x = 9x \\
&\text{Last: } 9 \cdot 6 = 54 \\
&\text{Combine: } x^2 + 6x + 9x + 54 = x^2 + 15x + 54
\end{aligned}
\]
Answer: \(x^2 + 15x + 54\)

##### Expression (8): \((x + 4)(x + 5)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 5 = 5x \\
&\text{Inner: } 4 \cdot x = 4x \\
&\text{Last: } 4 \cdot 5 = 20 \\
&\text{Combine: } x^2 + 5x + 4x + 20 = x^2 + 9x + 20
\end{aligned}
\]
Answer: \(x^2 + 9x + 20\)

##### Expression (9): \((x + 6)(x + 8)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 8 = 8x \\
&\text{Inner: } 6 \cdot x = 6x \\
&\text{Last: } 6 \cdot 8 = 48 \\
&\text{Combine: } x^2 + 8x + 6x + 48 = x^2 + 14x + 48
\end{aligned}
\]
Answer: \(x^2 + 14x + 48\)

##### Expression (10): \((x + 10)(x + 7)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 7 = 7x \\
&\text{Inner: } 10 \cdot x = 10x \\
&\text{Last: } 10 \cdot 7 = 70 \\
&\text{Combine: } x^2 + 7x + 10x + 70 = x^2 + 17x + 70
\end{aligned}
\]
Answer: \(x^2 + 17x + 70\)

##### Expression (11): \((x + 3)(x + 9)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 9 = 9x \\
&\text{Inner: } 3 \cdot x = 3x \\
&\text{Last: } 3 \cdot 9 = 27 \\
&\text{Combine: } x^2 + 9x + 3x + 27 = x^2 + 12x + 27
\end{aligned}
\]
Answer: \(x^2 + 12x + 27\)

##### Expression (12): \((x + 12)(x + 3)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 3 = 3x \\
&\text{Inner: } 12 \cdot x = 12x \\
&\text{Last: } 12 \cdot 3 = 36 \\
&\text{Combine: } x^2 + 3x + 12x + 36 = x^2 + 15x + 36
\end{aligned}
\]
Answer: \(x^2 + 15x + 36\)

##### Expression (13): \((x + 7)(x + 6)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 6 = 6x \\
&\text{Inner: } 7 \cdot x = 7x \\
&\text{Last: } 7 \cdot 6 = 42 \\
&\text{Combine: } x^2 + 6x + 7x + 42 = x^2 + 13x + 42
\end{aligned}
\]
Answer: \(x^2 + 13x + 42\)

##### Expression (14): \((x + 5)(x + 5)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 5 = 5x \\
&\text{Inner: } 5 \cdot x = 5x \\
&\text{Last: } 5 \cdot 5 = 25 \\
&\text{Combine: } x^2 + 5x + 5x + 25 = x^2 + 10x + 25
\end{aligned}
\]
Answer: \(x^2 + 10x + 25\)

##### Expression (15): \((x + 3)(x + 8)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 8 = 8x \\
&\text{Inner: } 3 \cdot x = 3x \\
&\text{Last: } 3 \cdot 8 = 24 \\
&\text{Combine: } x^2 + 8x + 3x + 24 = x^2 + 11x + 24
\end{aligned}
\]
Answer: \(x^2 + 11x + 24\)

##### Expression (16): \((x + 15)(x + 3)\)
\[
\begin{aligned}
&\text{First: } x \cdot x = x^2 \\
&\text{Outer: } x \cdot 3 = 3x \\
&\text{Inner: } 15 \cdot x = 15x \\
&\text{Last: } 15 \cdot 3 = 45 \\
&\text{Combine: } x^2 + 3x + 15x + 45 = x^2 + 18x + 45
\end{aligned}
\]
Answer: \(x^2 + 18x + 45\)

##### Expression (17): \((2x + 3)(x + 2)\)
\[
\begin{aligned}
&\text{First: } 2x \cdot x = 2x^2 \\
&\text{Outer: } 2x \cdot 2 = 4x \\
&\text{Inner: } 3 \cdot x = 3x \\
&\text{Last: } 3 \cdot 2 = 6 \\
&\text{Combine: } 2x^2 + 4x + 3x + 6 = 2x^2 + 7x + 6
\end{aligned}
\]
Answer: \(2x^2 + 7x + 6\)

##### Expression (18): \((3x + 1)(2x + 5)\)
\[
\begin{aligned}
&\text{First: } 3x \cdot 2x = 6x^2 \\
&\text{Outer: } 3x \cdot 5 = 15x \\
&\text{Inner: } 1 \cdot 2x = 2x \\
&\text{Last: } 1 \cdot 5 = 5 \\
&\text{Combine: } 6x^2 + 15x + 2x + 5 = 6x^2 + 17x + 5
\end{aligned}
\]
Answer: \(6x^2 + 17x + 5\)

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Final Answers:

\[
\boxed{
\begin{aligned}
&1. x^2 + 7x + 12 \\
&2. x^2 + 7x + 10 \\
&3. x^2 + 13x + 42 \\
&4. x^2 + 11x + 30 \\
&5. x^2 + 12x + 32 \\
&6. x^2 + 13x + 30 \\
&7. x^2 + 15x + 54 \\
&8. x^2 + 9x + 20 \\
&9. x^2 + 14x + 48 \\
&10. x^2 + 17x + 70 \\
&11. x^2 + 12x + 27 \\
&12. x^2 + 15x + 36 \\
&13. x^2 + 13x + 42 \\
&14. x^2 + 10x + 25 \\
&15. x^2 + 11x + 24 \\
&16. x^2 + 18x + 45 \\
&17. 2x^2 + 7x + 6 \\
&18. 6x^2 + 17x + 5
\end{aligned}
}
\]