To simplify the given polynomial expression:
[tex]\[
(5x^2 + 13x - 4) - (17x^2 + 7x - 19) + (5x - 7)(3x + 1)
\][/tex]
We will do this step-by-step:
1. Simplify each pair of parentheses separately.
- The first expression is:
[tex]\[
5x^2 + 13x - 4
\][/tex]
- The second expression is:
[tex]\[
-(17x^2 + 7x - 19) = -17x^2 - 7x + 19
\][/tex]
2. Combine these two expressions:
[tex]\[
(5x^2 + 13x - 4) + (-17x^2 - 7x + 19)
\][/tex]
Combine like terms:
[tex]\[
(5x^2 - 17x^2) + (13x - 7x) + (-4 + 19)
\][/tex]
[tex]\[
= -12x^2 + 6x + 15
\][/tex]
3. Simplify the product of the binomials (5x - 7) and (3x + 1):
[tex]\[
(5x - 7)(3x + 1)
\][/tex]
Distribute each term in the first binomial to each term in the second binomial:
[tex]\[
5x \cdot 3x + 5x \cdot 1 - 7 \cdot 3x - 7 \cdot 1
\][/tex]
[tex]\[
= 15x^2 + 5x - 21x - 7
\][/tex]
[tex]\[
= 15x^2 - 16x - 7
\][/tex]
4. Add the results of the combined terms and the simplified product of the binomials:
[tex]\[
(-12x^2 + 6x + 15) + (15x^2 - 16x - 7)
\][/tex]
Combine like terms again:
[tex]\[
(-12x^2 + 15x^2) + (6x - 16x) + (15 - 7)
\][/tex]
[tex]\[
= 3x^2 - 10x + 8
\][/tex]
Thus, the simplified polynomial expression can be written as:
[tex]\[
x^2 - 10x + 8
\][/tex]
So, filling in the boxes, we have:
[tex]\[
x^2 - \boxed{10} x + \boxed{8}
\][/tex]
