To solve the given polynomial expressions, we need to follow specific steps for each. I'll break down the process step-by-step.
### Expression 1: [tex]\((2x^2 + x + 3) + (3x^2 + 2x + 1)\)[/tex]
Step 1: Remove the parentheses
Combining the two expressions without changing any signs:
[tex]\[ (2x^2 + x + 3) + (3x^2 + 2x + 1) = 2x^2 + x + 3 + 3x^2 + 2x + 1 \][/tex]
Step 2: Collect like terms
We combine the terms with the same power of [tex]\( x \)[/tex]:
[tex]\[ 2x^2 + 3x^2 + x + 2x + 3 + 1 \][/tex]
Here, [tex]\(2x^2\)[/tex] and [tex]\(3x^2\)[/tex] are the [tex]\(x^2\)[/tex] terms, [tex]\(x\)[/tex] and [tex]\(2x\)[/tex] are the [tex]\(x\)[/tex] terms, and [tex]\(3\)[/tex] and [tex]\(1\)[/tex] are the constants.
Step 3: Simplify
Add the coefficients of like terms:
[tex]\[ (2 + 3)x^2 + (1 + 2)x + (3 + 1) \][/tex]
[tex]\[ 5x^2 + 3x + 4 \][/tex]
### Expression 2: [tex]\((x^3 - 2x + 7) + (-2x^3 + 5x - 1)\)[/tex]
Step 1: Remove the parentheses
Combining the two expressions without changing any signs:
[tex]\[ (x^3 - 2x + 7) + (-2x^3 + 5x - 1) = x^3 - 2x + 7 - 2x^3 + 5x - 1 \][/tex]
Step 2: Collect like terms
We combine the terms with the same power of [tex]\( x \)[/tex]:
[tex]\[ x^3 - 2x^3 - 2x + 5x + 7 - 1 \][/tex]
Here, [tex]\(x^3\)[/tex] and [tex]\(-2x^3\)[/tex] are the [tex]\(x^3\)[/tex] terms, [tex]\(-2x\)[/tex] and [tex]\(5x\)[/tex] are the [tex]\(x\)[/tex] terms, and [tex]\(7\)[/tex] and [tex]\(-1\)[/tex] are the constants.
Step 3: Simplify
Add the coefficients of like terms:
[tex]\[ (1 - 2)x^3 + (-2 + 5)x + (7 - 1) \][/tex]
[tex]\[ -x^3 + 3x + 6 \][/tex]
### Summary of Results:
- The simplified form of [tex]\((2x^2 + x + 3) + (3x^2 + 2x + 1)\)[/tex] is: [tex]\(\boxed{5x^2 + 3x + 4}\)[/tex]
- The simplified form of [tex]\((x^3 - 2x + 7) + (-2x^3 + 5x - 1)\)[/tex] is: [tex]\(\boxed{-x^3 + 3x + 6}\)[/tex]
These are the final simplified forms of the given polynomial expressions after following the detailed steps.
