To simplify the polynomial expression [tex]\((5x^2 + 13x - 4) - (17x^2 + 7x - 19) + (5x - 7)(3x + 1)\)[/tex], follow these steps:
1. Subtract the second polynomial from the first:
[tex]\[
(5x^2 + 13x - 4) - (17x^2 + 7x - 19)
\][/tex]
Distribute the negative sign:
[tex]\[
5x^2 + 13x - 4 - 17x^2 - 7x + 19
\][/tex]
Combine like terms:
[tex]\[
(5x^2 - 17x^2) + (13x - 7x) + (-4 + 19) = -12x^2 + 6x + 15
\][/tex]
2. Expand the product [tex]\((5x - 7)(3x + 1)\)[/tex]:
[tex]\[
(5x - 7)(3x + 1)
\][/tex]
Use the distributive property:
[tex]\[
5x \cdot 3x + 5x \cdot 1 - 7 \cdot 3x - 7 \cdot 1 = 15x^2 + 5x - 21x - 7
\][/tex]
Combine like terms:
[tex]\[
15x^2 + (5x - 21x) - 7 = 15x^2 - 16x - 7
\][/tex]
3. Add the results from the subtraction and the multiplication:
Combine [tex]\(-12x^2 + 6x + 15\)[/tex] and [tex]\(15x^2 - 16x - 7\)[/tex]:
[tex]\[
(-12x^2 + 6x + 15) + (15x^2 - 16x - 7)
\][/tex]
Combine like terms:
[tex]\[
(-12x^2 + 15x^2) + (6x - 16x) + (15 - 7) = 3x^2 - 10x + 8
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
3x^2 - 10x + 8
\][/tex]
Thus, the correct terms to fill in the boxes are:
[tex]\[
\boxed{3} x^2 - \boxed{10} x + \boxed{8}
\][/tex]
