To simplify the given polynomial expression:
[tex]\[
(5x^2 + 13x - 4) - (17x^2 + 7x - 19) + (5x - 7)(3x + 1)
\][/tex]
we follow these steps:
1. Distribute the negative sign in the second polynomial:
[tex]\[
(5x^2 + 13x - 4) + (-17x^2 - 7x + 19)
\][/tex]
2. Expand the product in the third term:
[tex]\[
(5x - 7)(3x + 1) = 5x \cdot 3x + 5x \cdot 1 - 7 \cdot 3x - 7 \cdot 1 = 15x^2 + 5x - 21x - 7 = 15x^2 - 16x - 7
\][/tex]
3. Combine all polynomials:
[tex]\[
(5x^2 + 13x - 4) + (-17x^2 - 7x + 19) + (15x^2 - 16x - 7)
\][/tex]
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 constant terms:
[tex]\[
-4 + 19 - 7 = 8
\][/tex]
Therefore, the simplified expression is:
[tex]\[
3x^2 - 10x + 8
\][/tex]
So, the correct answer in each box is:
[tex]\[
3 \, x^2 - 10 \, x + 8
\][/tex]
