To find the product of the given polynomials [tex]\( (5x + 5 - 2x)(4 + 7x - 1) \)[/tex], we should first combine like terms in each polynomial and then multiply the simplified forms. Here’s the step-by-step solution:
1. Simplify each polynomial:
[tex]\[
5x + 5 - 2x = (5x - 2x) + 5 = 3x + 5
\][/tex]
[tex]\[
4 + 7x - 1 = 7x + (4 - 1) = 7x + 3
\][/tex]
So, the given expression [tex]\( (5x + 5 - 2x)(4 + 7x - 1) \)[/tex] simplifies to [tex]\( (3x + 5)(7x + 3) \)[/tex].
2. Apply the distributive property (also known as the FOIL method for binomials) to multiply the polynomials:
[tex]\[
(3x + 5)(7x + 3) = 3x \cdot 7x + 3x \cdot 3 + 5 \cdot 7x + 5 \cdot 3
\][/tex]
3. Compute each product:
- [tex]\( 3x \cdot 7x = 21x^2 \)[/tex]
- [tex]\( 3x \cdot 3 = 9x \)[/tex]
- [tex]\( 5 \cdot 7x = 35x \)[/tex]
- [tex]\( 5 \cdot 3 = 15 \)[/tex]
4. Combine the results to form the polynomial expression:
[tex]\[
21x^2 + 9x + 35x + 15
\][/tex]
5. Combine like terms (the [tex]\( x \)[/tex]-terms):
[tex]\[
21x^2 + (9x + 35x) + 15 = 21x^2 + 44x + 15
\][/tex]
Thus, the product of the given polynomials [tex]\( (5x + 5 - 2x)(4 + 7x - 1) \)[/tex] simplifies to [tex]\( 21x^2 + 44x + 15 \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{21x^2 + 44x + 15}
\][/tex]
