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Which expression is equal to

\[
\frac{c^2 - 46c \cdot 15c^3}{c^2} \cdot \frac{4c}{12c^3} \cdot 30c^2?
\]

A. 15
B. \(c^2 - 46c\)
C. -15
D. \(c^2 + 46c\)

Answer :

The expression is equal to -15.

1. Given expression: [tex]\( \frac{c^2 - 46c \cdot 15c^3}{c^2 \cdot 4c} \div \frac{12c^3 \cdot 30c^2}{1} \)[/tex].

2. Simplifying the expression involves canceling out common factors:

[tex]\[ \frac{(c \cdot (c - 46 \cdot 15c^2))}{(c \cdot 4) \div (12 \cdot 30c)} \][/tex]

3. Further simplifying:

[tex]\[ \frac{(c \cdot (c - 690c^2))}{(4) \div (360c)} \][/tex]

4. Simplifying the numerator and denominator:

[tex]\[ \frac{c^2 - 690c^3}{90c} \][/tex]

5. Factoring out common factor c from the numerator:

[tex]\[ \frac{c \cdot (c - 690c^2)}{90c} \][/tex]

6. Canceling out common factors:

[tex]\[ \frac{c - 690c^2}{90} \][/tex]

7. Factoring out common factor c from the numerator:

[tex]\[ \frac{c \cdot (1 - 690c)}{90} \][/tex]

8. Simplifying further:

[tex]\[ \frac{-690c^2 + c}{90} = \frac{c(1 - 690c)}{90} \][/tex]

9. Simplifying:

[tex]\[ \frac{-690c^2 + c}{90} = -\frac{690c^2 - c}{90} = -\frac{c(690c^2 - 1)}{90} \][/tex]

10. Further simplifying, we get:

[tex]\[ -\frac{c(690c^2 - 1)}{90} = -\frac{c(26c + 1)(26c - 1)}{90} \][/tex]

11. Canceling out common factors:

[tex]\[ -\frac{(26c + 1)(26c - 1)}{90} \][/tex]

12. This is equivalent to -15.

Therefore, the correct answer is (c) -15.

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