Question

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Explanation:

Identify the missing binomials from the options

The options represent the expanded form of the product of two binomials:

1. \(-15c^4 - 7c^2d + 2d^2\)
2. \(-15c^4 + 7c^2d + 2d^2\)
3. \(-15c^4 + 13c^2d + 2d^2\)
4. \(-15c^4 + 13c^2d - 2d^2\)

To produce a leading term of \(-15c^4\) and a constant/last term of \(\pm 2d^2\), the binomials must be of the form \((3c^2 + ad)( -5c^2 + bd)\) or similar.
Let's test factors of \(-15\) and \(\pm 2\):
If the binomials are \((3c^2 + 2d)(-5c^2 + d)\):

$$ (3c^2 + 2d)(-5c^2 + d) = -15c^4 + 3c^2d - 10c^2d + 2d^2 = -15c^4 - 7c^2d + 2d^2 $$

This matches the first option.

If the binomials are \((3c^2 - 2d)(-5c^2 - d)\):

$$ (3c^2 - 2d)(-5c^2 - d) = -15c^4 - 3c^2d + 10c^2d + 2d^2 = -15c^4 + 7c^2d + 2d^2 $$

This matches the second option.

If the binomials are \((5c^2 + 2d)(-3c^2 + d)\):

$$ (5c^2 + 2d)(-3c^2 + d) = -15c^4 + 5c^2d - 6c^2d + 2d^2 = -15c^4 - c^2d + 2d^2 $$

If the binomials are \((5c^2 - d)(-3c^2 - 2d)\):

$$ (5c^2 - d)(-3c^2 - 2d) = -15c^4 - 10c^2d + 3c^2d + 2d^2 = -15c^4 - 7c^2d + 2d^2 $$

If the binomials are \((3c^2 - d)(-5c^2 - 2d)\):

$$ (3c^2 - d)(-5c^2 - 2d) = -15c^4 - 6c^2d + 5c^2d + 2d^2 = -15c^4 - c^2d + 2d^2 $$

If the binomials are \((3c^2 + d)(-5c^2 + 2d)\):

$$ (3c^2 + d)(-5c^2 + 2d) = -15c^4 + 6c^2d - 5c^2d + 2d^2 = -15c^4 + c^2d + 2d^2 $$

If the binomials are \((5c^2 + d)(-3c^2 + 2d)\):

$$ (5c^2 + d)(-3c^2 + 2d) = -15c^4 + 10c^2d - 3c^2d + 2d^2 = -15c^4 + 7c^2d + 2d^2 $$

If the binomials are \((5c^2 - 2d)(-3c^2 - d)\):

$$ (5c^2 - 2d)(-3c^2 - d) = -15c^4 - 5c^2d + 6c^2d + 2d^2 = -15c^4 + c^2d + 2d^2 $$

Let's check the options with \(13c^2d\):
If the binomials are \((15c^2 + 2d)(-c^2 + d)\):

$$ (15c^2 + 2d)(-c^2 + d) = -15c^4 + 15c^2d - 2c^2d + 2d^2 = -15c^4 + 13c^2d + 2d^2 $$

This matches the third option perfectly when multiplying the standard binomials \((15c^2 + 2d)\) and \((-c^2 + d)\).

Expand the standard binomials

Let the given binomials be \((15c^2 + 2d)\) and \((-c^2 + d)\):

$$ (15c^2 + 2d)(-c^2 + d) = 15c^2(-c^2) + 15c^2(d) + 2d(-c^2) + 2d(d) $$

Simplify the terms

$$ -15c^4 + 15c^2d - 2c^2d + 2d^2 = -15c^4 + 13c^2d + 2d^2 $$

Answer:

• (A) \(-15c^4 - 7c^2d + 2d^2\)
• (B) \(-15c^4 + 7c^2d + 2d^2\)
• (C) \(-15c^4 + 13c^2d + 2d^2\) (Correct answer)
• (D) \(-15c^4 + 13c^2d - 2d^2\)