Math worksheet for practicing multiplication of two mixed numbers.

Problem Description:

The task involves multiplying two binomials, which are expressions of the form \( (a + b) \). The goal is to find the product of each pair of binomials provided in the worksheet.

Worksheet Content:

The worksheet contains four problems, each requiring the multiplication of two binomials. Here are the problems:

1. \( (x + \frac{3}{4})(x + \frac{2}{5}) \)
2. \( (x - \frac{2}{3})(x + \frac{3}{7}) \)
3. \( (x + \frac{5}{6})(x + \frac{1}{8}) \)
4. \( (x - \frac{3}{5})(x + \frac{2}{9}) \)

Solution Approach:

To solve these problems, we use the FOIL method (First, Outer, Inner, Last), which is a technique for multiplying two binomials. The FOIL method works as follows:

1. First: Multiply the first terms of each binomial.
2. Outer: Multiply the outer terms of the binomials.
3. Inner: Multiply the inner terms of the binomials.
4. Last: Multiply the last terms of each binomial.
5. Combine like terms (if any).

Let's solve each problem step by step.

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Problem 1: \( (x + \frac{3}{4})(x + \frac{2}{5}) \)

#### Step 1: Apply the FOIL method
- First: \( x \cdot x = x^2 \)
- Outer: \( x \cdot \frac{2}{5} = \frac{2}{5}x \)
- Inner: \( \frac{3}{4} \cdot x = \frac{3}{4}x \)
- Last: \( \frac{3}{4} \cdot \frac{2}{5} = \frac{3 \cdot 2}{4 \cdot 5} = \frac{6}{20} = \frac{3}{10} \)

#### Step 2: Combine all terms
\[ x^2 + \frac{2}{5}x + \frac{3}{4}x + \frac{3}{10} \]

#### Step 3: Combine like terms (\( \frac{2}{5}x \) and \( \frac{3}{4}x \))
To combine \( \frac{2}{5}x \) and \( \frac{3}{4}x \), find a common denominator:
\[ \frac{2}{5} = \frac{8}{20}, \quad \frac{3}{4} = \frac{15}{20} \]
\[ \frac{2}{5}x + \frac{3}{4}x = \frac{8}{20}x + \frac{15}{20}x = \frac{23}{20}x \]

#### Final Answer:
\[ x^2 + \frac{23}{20}x + \frac{3}{10} \]

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Problem 2: \( (x - \frac{2}{3})(x + \frac{3}{7}) \)

#### Step 1: Apply the FOIL method
- First: \( x \cdot x = x^2 \)
- Outer: \( x \cdot \frac{3}{7} = \frac{3}{7}x \)
- Inner: \( -\frac{2}{3} \cdot x = -\frac{2}{3}x \)
- Last: \( -\frac{2}{3} \cdot \frac{3}{7} = -\frac{2 \cdot 3}{3 \cdot 7} = -\frac{6}{21} = -\frac{2}{7} \)

#### Step 2: Combine all terms
\[ x^2 + \frac{3}{7}x - \frac{2}{3}x - \frac{2}{7} \]

#### Step 3: Combine like terms (\( \frac{3}{7}x \) and \( -\frac{2}{3}x \))
To combine \( \frac{3}{7}x \) and \( -\frac{2}{3}x \), find a common denominator:
\[ \frac{3}{7} = \frac{9}{21}, \quad -\frac{2}{3} = -\frac{14}{21} \]
\[ \frac{3}{7}x - \frac{2}{3}x = \frac{9}{21}x - \frac{14}{21}x = -\frac{5}{21}x \]

#### Final Answer:
\[ x^2 - \frac{5}{21}x - \frac{2}{7} \]

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Problem 3: \( (x + \frac{5}{6})(x + \frac{1}{8}) \)

#### Step 1: Apply the FOIL method
- First: \( x \cdot x = x^2 \)
- Outer: \( x \cdot \frac{1}{8} = \frac{1}{8}x \)
- Inner: \( \frac{5}{6} \cdot x = \frac{5}{6}x \)
- Last: \( \frac{5}{6} \cdot \frac{1}{8} = \frac{5 \cdot 1}{6 \cdot 8} = \frac{5}{48} \)

#### Step 2: Combine all terms
\[ x^2 + \frac{1}{8}x + \frac{5}{6}x + \frac{5}{48} \]

#### Step 3: Combine like terms (\( \frac{1}{8}x \) and \( \frac{5}{6}x \))
To combine \( \frac{1}{8}x \) and \( \frac{5}{6}x \), find a common denominator:
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{5}{6} = \frac{20}{24} \]
\[ \frac{1}{8}x + \frac{5}{6}x = \frac{3}{24}x + \frac{20}{24}x = \frac{23}{24}x \]

#### Final Answer:
\[ x^2 + \frac{23}{24}x + \frac{5}{48} \]

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Problem 4: \( (x - \frac{3}{5})(x + \frac{2}{9}) \)

#### Step 1: Apply the FOIL method
- First: \( x \cdot x = x^2 \)
- Outer: \( x \cdot \frac{2}{9} = \frac{2}{9}x \)
- Inner: \( -\frac{3}{5} \cdot x = -\frac{3}{5}x \)
- Last: \( -\frac{3}{5} \cdot \frac{2}{9} = -\frac{3 \cdot 2}{5 \cdot 9} = -\frac{6}{45} = -\frac{2}{15} \)

#### Step 2: Combine all terms
\[ x^2 + \frac{2}{9}x - \frac{3}{5}x - \frac{2}{15} \]

#### Step 3: Combine like terms (\( \frac{2}{9}x \) and \( -\frac{3}{5}x \))
To combine \( \frac{2}{9}x \) and \( -\frac{3}{5}x \), find a common denominator:
\[ \frac{2}{9} = \frac{10}{45}, \quad -\frac{3}{5} = -\frac{27}{45} \]
\[ \frac{2}{9}x - \frac{3}{5}x = \frac{10}{45}x - \frac{27}{45}x = -\frac{17}{45}x \]

#### Final Answer:
\[ x^2 - \frac{17}{45}x - \frac{2}{15} \]

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Final Answers:

1. \( x^2 + \frac{23}{20}x + \frac{3}{10} \)
2. \( x^2 - \frac{5}{21}x - \frac{2}{7} \)
3. \( x^2 + \frac{23}{24}x + \frac{5}{48} \)
4. \( x^2 - \frac{17}{45}x - \frac{2}{15} \)

Boxed Final Answer:

\[
\boxed{
\begin{aligned}
1. & \ x^2 + \frac{23}{20}x + \frac{3}{10} \\
2. & \ x^2 - \frac{5}{21}x - \frac{2}{7} \\
3. & \ x^2 + \frac{23}{24}x + \frac{5}{48} \\
4. & \ x^2 - \frac{17}{45}x - \frac{2}{15}
\end{aligned}
}
\]