Answer :
To find the product [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we need to use the distributive property, often referred to as the FOIL method when dealing with binomials. Let's go through the process step-by-step:
1. Distribute each term in the first binomial to each term in the second binomial:
[tex]\[
(-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3)
\][/tex]
2. Calculate each of these products one by one:
- First Term: [tex]\((-2x)(-4x) = 8x^2\)[/tex]
(Multiplying the coefficients [tex]\(-2\)[/tex] and [tex]\(-4\)[/tex] gives [tex]\(8\)[/tex], and [tex]\(x \cdot x = x^2\)[/tex])
- Second Term: [tex]\((-2x)(-3) = 6x\)[/tex]
(Multiplying the coefficients [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex] gives [tex]\(6\)[/tex] and considering the x, the term is [tex]\(6x\)[/tex])
- Third Term: [tex]\((-9y^2)(-4x) = 36xy^2\)[/tex]
(Multiplying the coefficients [tex]\(-9\)[/tex] and [tex]\(-4\)[/tex] gives [tex]\(36\)[/tex], [tex]\(y^2\)[/tex] stays as is, and then multiply by [tex]\(x\)[/tex] to give [tex]\(36xy^2\)[/tex])
- Fourth Term: [tex]\((-9y^2)(-3) = 27y^2\)[/tex]
(Multiplying the coefficients [tex]\(-9\)[/tex] and [tex]\(-3\)[/tex] gives [tex]\(27\)[/tex] and keeping [tex]\(y^2\)[/tex])
3. Combine all the terms together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
The final expanded expression is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
So, the correct answer from the options given is: [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
1. Distribute each term in the first binomial to each term in the second binomial:
[tex]\[
(-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3)
\][/tex]
2. Calculate each of these products one by one:
- First Term: [tex]\((-2x)(-4x) = 8x^2\)[/tex]
(Multiplying the coefficients [tex]\(-2\)[/tex] and [tex]\(-4\)[/tex] gives [tex]\(8\)[/tex], and [tex]\(x \cdot x = x^2\)[/tex])
- Second Term: [tex]\((-2x)(-3) = 6x\)[/tex]
(Multiplying the coefficients [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex] gives [tex]\(6\)[/tex] and considering the x, the term is [tex]\(6x\)[/tex])
- Third Term: [tex]\((-9y^2)(-4x) = 36xy^2\)[/tex]
(Multiplying the coefficients [tex]\(-9\)[/tex] and [tex]\(-4\)[/tex] gives [tex]\(36\)[/tex], [tex]\(y^2\)[/tex] stays as is, and then multiply by [tex]\(x\)[/tex] to give [tex]\(36xy^2\)[/tex])
- Fourth Term: [tex]\((-9y^2)(-3) = 27y^2\)[/tex]
(Multiplying the coefficients [tex]\(-9\)[/tex] and [tex]\(-3\)[/tex] gives [tex]\(27\)[/tex] and keeping [tex]\(y^2\)[/tex])
3. Combine all the terms together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
The final expanded expression is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
So, the correct answer from the options given is: [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
