To find the product of the given polynomials [tex]\( (5x + 5 - 2x)(4 + 7x - 1) \)[/tex], we need to multiply every term in the first polynomial by every term in the second polynomial. Let's carefully go through the multiplication step-by-step:
1. First, rewrite the polynomials in a standard form:
[tex]\[ (3x + 5)(4 + 7x - 1) \][/tex]
(the term [tex]\(5x - 2x\)[/tex] combines to [tex]\(3x\)[/tex]).
2. Distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[
(3x + 5 - 2x)(4 + 7x - 1) = (3x + 5)(4 + 7x - 1)
\][/tex]
3. Distribute [tex]\(5x\)[/tex]:
- [tex]\(5x \cdot 4 = 20x\)[/tex]
- [tex]\(5x \cdot 7x = 35x^2\)[/tex]
- [tex]\(5x \cdot (-1) = -5x\)[/tex]
4. Distribute [tex]\(5\)[/tex]:
- [tex]\(5 \cdot 4 = 20\)[/tex]
- [tex]\(5 \cdot 7x = 35x\)[/tex]
- [tex]\(5 \cdot (-1) = -5\)[/tex]
5. Distribute [tex]\(-2x\)[/tex]:
- [tex]\(-2x \cdot 4 = -8x\)[/tex]
- [tex]\(-2x \cdot 7x = -14x^2\)[/tex]
- [tex]\(-2x \cdot (-1) = 2x\)[/tex]
6. Combine all these terms, we get:
[tex]\[
20x + 35x^2 - 5x + 20 + 35x - 5 - 8x - 14x^2 + 2x
= (35x^2 - 14x^2) + (20x - 8x + 2x+35x-5x ) + (20 - 5)
= 21x^2 + 24x + 15
\][/tex]
So, the product of the polynomials is:
[tex]\[ \boxed{21x^2 + 24x + 15} \][/tex]
Therefore, the given expression evaluates to [tex]\(21x^2 + 24x + 15\)[/tex] which matches the answer among the provided ones.
