Certainly! To multiply the expressions [tex]\((5 + 2x)\)[/tex] and [tex]\((4 - 7x)\)[/tex] step-by-step, we will use the distributive property. The distributive property states that each term in the first expression must be multiplied by each term in the second expression. Let's follow the steps carefully:
1. Distribute the terms:
- Multiply the constant term [tex]\(5\)[/tex] from the first expression by each term in the second expression.
- Multiply the term [tex]\(2x\)[/tex] from the first expression by each term in the second expression.
2. Perform the multiplications:
- [tex]\(5 \times 4 = 20\)[/tex]
- [tex]\(5 \times (-7x) = -35x\)[/tex]
- [tex]\(2x \times 4 = 8x\)[/tex]
- [tex]\(2x \times (-7x) = -14x^2\)[/tex]
3. Combine the results:
- The constant term: [tex]\(20\)[/tex]
- The linear terms: [tex]\(-35x + 8x\)[/tex]
- The quadratic term: [tex]\(-14x^2\)[/tex]
4. Add the linear terms:
- Combine [tex]\(-35x\)[/tex] and [tex]\(8x\)[/tex] to get [tex]\(-27x\)[/tex]
5. Write the simplified expression:
[tex]\[
20 - 27x - 14x^2
\][/tex]
6. Organize the terms in standard form (highest power of [tex]\(x\)[/tex] first):
[tex]\[
-14x^2 - 27x + 20
\][/tex]
Thus, the multiplication and simplification of [tex]\((5 + 2x)\)[/tex] and [tex]\((4 - 7x)\)[/tex] results in the expression:
[tex]\[
-14x^2 - 27x + 20
\][/tex]
