Answer :
To multiply the polynomials
$$\left(8x^2 + 6x + 8\right)(6x - 5),$$
we use the distributive property (also known as the FOIL method for two binomials, but here we extend it as needed).
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply $8x^2$ by $6x$:
$$8x^2 \cdot 6x = 48x^3.$$
- Multiply $8x^2$ by $-5$:
$$8x^2 \cdot (-5) = -40x^2.$$
- Multiply $6x$ by $6x$:
$$6x \cdot 6x = 36x^2.$$
- Multiply $6x$ by $-5$:
$$6x \cdot (-5) = -30x.$$
- Multiply $8$ by $6x$:
$$8 \cdot 6x = 48x.$$
- Multiply $8$ by $-5$:
$$8 \cdot (-5) = -40.$$
2. Now, combine the like terms:
- The only $x^3$ term is:
$$48x^3.$$
- Combine the $x^2$ terms:
$$-40x^2 + 36x^2 = -4x^2.$$
- Combine the $x$ terms:
$$-30x + 48x = 18x.$$
- The constant term is:
$$-40.$$
3. Write the final simplified polynomial by putting the terms together:
$$48x^3 - 4x^2 + 18x - 40.$$
This matches the expression in Option C.
$$\left(8x^2 + 6x + 8\right)(6x - 5),$$
we use the distributive property (also known as the FOIL method for two binomials, but here we extend it as needed).
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply $8x^2$ by $6x$:
$$8x^2 \cdot 6x = 48x^3.$$
- Multiply $8x^2$ by $-5$:
$$8x^2 \cdot (-5) = -40x^2.$$
- Multiply $6x$ by $6x$:
$$6x \cdot 6x = 36x^2.$$
- Multiply $6x$ by $-5$:
$$6x \cdot (-5) = -30x.$$
- Multiply $8$ by $6x$:
$$8 \cdot 6x = 48x.$$
- Multiply $8$ by $-5$:
$$8 \cdot (-5) = -40.$$
2. Now, combine the like terms:
- The only $x^3$ term is:
$$48x^3.$$
- Combine the $x^2$ terms:
$$-40x^2 + 36x^2 = -4x^2.$$
- Combine the $x$ terms:
$$-30x + 48x = 18x.$$
- The constant term is:
$$-40.$$
3. Write the final simplified polynomial by putting the terms together:
$$48x^3 - 4x^2 + 18x - 40.$$
This matches the expression in Option C.
