Let's simplify the given polynomial expression step by step:
[tex]\[
\left(5x^2 + 13x - 4\right) - \left(17x^2 + 7x - 19\right) + (5x - 7)(3x + 1)
\][/tex]
### Step 1: Distribute the negative sign through the second polynomial
[tex]\[
\left(5x^2 + 13x - 4\right) - 17x^2 - 7x + 19 + (5x - 7)(3x + 1)
\][/tex]
### Step 2: Expand the third polynomial expression [tex]\((5x - 7)(3x + 1)\)[/tex]
[tex]\[
(5x - 7)(3x + 1) = 5x \cdot 3x + 5x \cdot 1 - 7 \cdot 3x - 7 \cdot 1
\][/tex]
[tex]\[
= 15x^2 + 5x - 21x - 7
\][/tex]
[tex]\[
= 15x^2 - 16x - 7
\][/tex]
### Step 3: Combine all the expressions together
[tex]\[
5x^2 + 13x - 4 - 17x^2 - 7x + 19 + 15x^2 - 16x - 7
\][/tex]
### Step 4: Combine like terms
- For [tex]\(x^2\)[/tex] terms:
[tex]\[
5x^2 - 17x^2 + 15x^2 = 3x^2
\][/tex]
- For [tex]\(x\)[/tex] terms:
[tex]\[
13x - 7x - 16x = -10x
\][/tex]
- For the constant terms:
[tex]\[
-4 + 19 - 7 = 8
\][/tex]
### Step 5: Write the simplified polynomial expression
[tex]\[
3x^2 - 10x + 8
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
\boxed{3x^2 - 10x + 8}
\][/tex]
