Answer :
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]
[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]
