`(ax+3)(5x^2 -bx+4)=20x^3 -9x^2 -2x +12`
The equation above is true for all x, where a and b are constants. What is the value of ab ?

Options

A. 18
B. 20
C. 24
D. 40

Explanation

To solve the equation \((ax + 3)(5x^2 - bx + 4) = 20x^3 - 9x^2 - 2x + 12\) and find the value of \(ab\), we will follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the left-hand side of the equation:

\[ (ax + 3)(5x^2 - bx + 4) \]

Using the distributive property (also known as the FOIL method for binomials), we can expand this:

\[ = ax(5x^2) + ax(-bx) + ax(4) + 3(5x^2) + 3(-bx) + 3(4) \]

Calculating each term:

- \(ax(5x^2) = 5ax^3\)
- \(ax(-bx) = -abx^2\)
- \(ax(4) = 4ax\)
- \(3(5x^2) = 15x^2\)
- \(3(-bx) = -3bx\)
- \(3(4) = 12\)

Combining these, we have:

\[ 5ax^3 + (-ab + 15)x^2 + (4a - 3b)x + 12 \]

### Step 2: Set the expanded form equal to the right-hand side
Now we set this equal to the right-hand side of the original equation:

\[ 5ax^3 + (-ab + 15)x^2 + (4a - 3b)x + 12 = 20x^3 - 9x^2 - 2x + 12 \]

### Step 3: Compare coefficients
Next, we compare the coefficients of the corresponding powers of \(x\) from both sides:

1. Coefficient of \(x^3\):
\[ 5a = 20 \implies a = \frac{20}{5} = 4 \]

2. Coefficient of \(x^2\):
\[ -ab + 15 = -9 \implies -ab = -9 - 15 \implies -ab = -24 \implies ab = 24 \]

3. Coefficient of \(x\):
\[ 4a - 3b = -2 \]

### Step 4: Solve for \(b\)
We already found \(a = 4\). Now we substitute \(a\) into the equation:

\[ 4(4) - 3b = -2 \implies 16 - 3b = -2 \implies -3b = -2 - 16 \implies -3b = -18 \implies b = \frac{-18}{-3} = 6 \]

### Step 5: Calculate \(ab\)
Now that we have \(a = 4\) and \(b = 6\):

\[ ab = 4 \times 6 = 24 \]

### Final Answer
Thus, the value of \(ab\) is:

\[ \boxed{24} \]