Let's simplify each of the given polynomial expressions step-by-step:
### 1. Simplifying [tex]\((5x - 3) + (2x + 4)\)[/tex]
Combine like terms:
- Combine the [tex]\(x\)[/tex] terms: [tex]\(5x + 2x = 7x\)[/tex]
- Combine the constant terms: [tex]\(-3 + 4 = 1\)[/tex]
So, the simplified expression is:
[tex]\[ \boxed{7x + 1} \][/tex]
### 2. Simplifying [tex]\((3x - 7) - (5x - 3)\)[/tex]
Distribute the negative sign across the second polynomial and then combine like terms:
- Distribute the negative sign: [tex]\( (3x - 7) - (5x - 3) = 3x - 7 - 5x + 3 \)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(3x - 5x = -2x\)[/tex]
- Combine the constant terms: [tex]\(-7 + 3 = -4\)[/tex]
So, the simplified expression is:
[tex]\[ \boxed{-2x - 4} \][/tex]
### 3. Simplifying [tex]\(7(2x + 4) + 6x\)[/tex]
Distribute the constant 7 through the terms inside the parentheses and then combine like terms:
- Distribute the 7: [tex]\( 7 \cdot 2x + 7 \cdot 4 = 14x + 28 \)[/tex]
- Add the remaining [tex]\(6x\)[/tex]: [tex]\( 14x + 28 + 6x \)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(14x + 6x = 20x\)[/tex]
So, the simplified expression is:
[tex]\[ \boxed{20x + 28} \][/tex]
### 4. Simplifying [tex]\(-6x(2x + 4) + (-5x^2 - 4x - 8)\)[/tex]
First, distribute [tex]\(-6x\)[/tex] through the terms inside the parentheses and then combine like terms:
- Distribute [tex]\(-6x\)[/tex]: [tex]\(-6x \cdot 2x + (-6x) \cdot 4 = -12x^2 - 24x\)[/tex]
- Add the resulting polynomials:
[tex]\[ -12x^2 - 24x + (-5x^2 - 4x - 8) \][/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-12x^2 - 5x^2 = -17x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-24x - 4x = -28x\)[/tex]
- The constant term remains the same: [tex]\(-8\)[/tex]
So, the simplified expression is:
[tex]\[ \boxed{-17x^2 - 28x - 8} \][/tex]
Here is the completed table with the original and simplified expressions:
[tex]\[
\begin{tabular}{||c|c||}
\hline Original Expression & Simplified Expression \\
\hline \hline
$(5 x-3)+(2 x+4)$ & $7x + 1$ \\
\hline
$(3 x-7)-(5 x-3)$ & $-2x - 4$ \\
\hline
$7(2 x+4)+6 x$ & $20x + 28$ \\
\hline
$-6 x(2 x+4)+\left(-5 x^2-4 x-8\right)$ & $-17x^2 - 28x - 8$ \\
\hline
\end{tabular}
\][/tex]
