High School

What is the product of the expression?

[tex](-2x - 9y^2)(-4x - 3)[/tex]

A. [tex]-8x^2 - 6x - 36xy^2 - 27y^2[/tex]

B. [tex]-14x^2 - 36xy^2 + 27y^2[/tex]

C. [tex]8x^2 + 6x + 36xy^2 + 27y^2[/tex]

D. [tex]14x^2 + 36xy^2 + 27y^2[/tex]

Answer :

Sure! Let's break down the multiplication step by step:

We want to multiply two expressions: [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex].

1. Distribute [tex]\((-4x)\)[/tex] across the first expression [tex]\((-2x - 9y^2)\)[/tex]:
- Multiply [tex]\(-4x\)[/tex] by [tex]\(-2x\)[/tex]: [tex]\((-4x) \times (-2x) = 8x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] by [tex]\(-9y^2\)[/tex]: [tex]\((-4x) \times (-9y^2) = 36xy^2\)[/tex]

2. Distribute [tex]\((-3)\)[/tex] across the first expression [tex]\((-2x - 9y^2)\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by [tex]\(-2x\)[/tex]: [tex]\((-3) \times (-2x) = 6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by [tex]\(-9y^2\)[/tex]: [tex]\((-3) \times (-9y^2) = 27y^2\)[/tex]

3. Combine all the terms:
- Collect all the results from steps 1 and 2: [tex]\(8x^2 + 36xy^2 + 6x + 27y^2\)[/tex]

The expanded expression, after performing all the multiplications, is:

[tex]\[ 8x^2 + 36xy^2 + 6x + 27y^2 \][/tex]

This matches one of the options provided:

[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]

Therefore, the product of [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex] is [tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex].