To solve the expression [tex]\((2x^2 + 24x + 1)(x + 3)\)[/tex] by expanding it, we need to distribute each term in the first polynomial [tex]\((2x^2 + 24x + 1)\)[/tex] across the second polynomial [tex]\((x + 3)\)[/tex]. Here's a detailed step-by-step method:
### Step-by-Step Solution:
1. Distribute [tex]\(2x^2\)[/tex]:
- Multiply [tex]\(2x^2\)[/tex] by each term in [tex]\((x + 3)\)[/tex]:
[tex]\[
2x^2 \cdot x = 2x^3
\][/tex]
[tex]\[
2x^2 \cdot 3 = 6x^2
\][/tex]
2. Distribute [tex]\(24x\)[/tex]:
- Multiply [tex]\(24x\)[/tex] by each term in [tex]\((x + 3)\)[/tex]:
[tex]\[
24x \cdot x = 24x^2
\][/tex]
[tex]\[
24x \cdot 3 = 72x
\][/tex]
3. Distribute [tex]\(1\)[/tex]:
- Multiply [tex]\(1\)[/tex] by each term in [tex]\((x + 3)\)[/tex]:
[tex]\[
1 \cdot x = x
\][/tex]
[tex]\[
1 \cdot 3 = 3
\][/tex]
4. Combine all the expanded terms:
- Now, let's sum up all the terms obtained from the above distributions:
[tex]\[
2x^3 + 6x^2 + 24x^2 + 72x + x + 3
\][/tex]
5. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
6x^2 + 24x^2 = 30x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
72x + x = 73x
\][/tex]
6. Write the final expanded form:
[tex]\[
2x^3 + 30x^2 + 73x + 3
\][/tex]
Thus, the expanded form of the given expression [tex]\((2x^2 + 24x + 1)(x + 3)\)[/tex] is:
[tex]\[
\boxed{2x^3 + 30x^2 + 73x + 3}
\][/tex]
