Final answer:
To find the product of the trinomials (7m²-8m-1)(8m²+7m+2), each term of the first trinomial is multiplied by each term of the second, and the resulting products are combined. The final product is 56m⁴ - 15m³ - 50m² - 23m - 2.
Step-by-step explanation:
The task is to find the product of two trinomials: (7m²-8m-1) and (8m²+7m+2). To do this, each term in the first trinomial is multiplied by each term in the second trinomial, and the results are then combined. This method is often referred to as the FOIL method (First, Outer, Inner, Last), although in the case of trinomials with more than two terms per polynomial, it is simply an extended version of distributive multiplication. The process involves multiplying term by term and then adding the products, making sure to combine like terms.
Starting with the multiplication:
- (7m² × 8m²) results in 56m⁴.
- (7m² × 7m) results in 49m³.
- (7m² × 2) results in 14m².
- (-8m × 8m²) results in -64m³.
- (-8m × 7m) results in -56m².
- (-8m × 2) results in -16m.
- (-1 × 8m²) results in -8m².
- (-1 × 7m) results in -7m.
- (-1 × 2) results in -2.
After multiplying, the like terms are combined:
- The m⁴ terms: 56m⁴.
- The m³ terms: 49m³ - 64m³ = -15m³.
- The m² terms: 14m² - 56m² - 8m² = -50m².
- The m terms: -16m - 7m = -23m.
- The constant terms: -2.
Thus, the product of the trinomials is 56m⁴ - 15m³ - 50m² - 23m - 2.