Question

which of the following is equivalent to $40a^2b - 24ab^2 - 16a$?
options:

• $-4a(-10ab + 6b^2 + 4)$
• $2ab(20a - 12b - 8)$
• $4ab(20a - 12b - 4b)$
• $-4b(-10a^2 + 6ab - 16a)$

Explanation:

Step1: Analyze each option

We need to expand each option and check if it equals \(40a^{2}b - 24ab^{2}-16a\).

Step2: Expand Option 1: \(-4a(-10ab + 6b^{2}+4)\)

Using the distributive property \(c(a + b + d)=ca+cb+cd\), we have:
\(-4a\times(-10ab)+(-4a)\times6b^{2}+(-4a)\times4\)
\( = 40a^{2}b-24ab^{2}-16a\)
Which matches the given expression. Let's check other options for confirmation.

Step3: Expand Option 2: \(2ab(20a - 12b - 8)\)

\(2ab\times20a+2ab\times(-12b)+2ab\times(-8)\)
\( = 40a^{2}b-24ab^{2}-16ab\) (Does not match)

Step4: Expand Option 3: \(4ab(20a - 12b - 4b)\)

Simplify the bracket first: \(20a-16b\)
Then expand: \(4ab\times20a+4ab\times(-16b)\)
\( = 80a^{2}b-64ab^{2}\) (Does not match)

Step5: Expand Option 4: \(-4b(-10a^{2}+6ab - 16a)\)

\(-4b\times(-10a^{2})+(-4b)\times6ab+(-4b)\times(-16a)\)
\( = 40a^{2}b-24ab^{2}+64ab\) (Does not match)

Answer:

\(-4a(-10ab + 6b^{2}+4)\) (The first option in the list)