College

What is the product of [tex]\left(2x^2 + 3x - 1\right)[/tex] and [tex](3x + 5)[/tex]?

A. [tex]6x^3 + 19x^2 - 12x + 5[/tex]
B. [tex]6x^3 + 9x^2 - 3x - 5[/tex]
C. [tex]6x^3 + 19x^2 + 12x - 5[/tex]
D. [tex]6x^3 + 10x^2 + 15x - 5[/tex]

Answer :

To find the product of
[tex]$$
(2x^2 + 3x - 1) \quad \text{and} \quad (3x + 5),
$$[/tex]
we multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

Step 1: Multiply the terms

1. Multiply [tex]$2x^2$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
2x^2 \cdot 3x = 6x^3,
$$[/tex]
[tex]$$
2x^2 \cdot 5 = 10x^2.
$$[/tex]

2. Multiply [tex]$3x$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
3x \cdot 3x = 9x^2,
$$[/tex]
[tex]$$
3x \cdot 5 = 15x.
$$[/tex]

3. Multiply [tex]$-1$[/tex] by each term in [tex]$(3x + 5)$[/tex]:
[tex]$$
-1 \cdot 3x = -3x,
$$[/tex]
[tex]$$
-1 \cdot 5 = -5.
$$[/tex]

Step 2: Write the expression with all terms

Combining all the terms together, we have:
[tex]$$
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5.
$$[/tex]

Step 3: Combine like terms

1. There is only one [tex]$x^3$[/tex] term:
[tex]$$
6x^3.
$$[/tex]

2. Combine the [tex]$x^2$[/tex] terms:
[tex]$$
10x^2 + 9x^2 = 19x^2.
$$[/tex]

3. Combine the [tex]$x$[/tex] terms:
[tex]$$
15x - 3x = 12x.
$$[/tex]

4. The constant term remains:
[tex]$$
-5.
$$[/tex]

Thus, the product is:
[tex]$$
6x^3 + 19x^2 + 12x - 5.
$$[/tex]

Final Answer: Option C.