Question

Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, squared, minus, 5, x, plus, 1, right parenthesis, left parenthesis, 3, x, squared, minus, x, minus, 4, right parenthesis
(2x
2
−5x+1)(3x
2
−x−4)

Answer 1

To expand the expression \((2x^2 - 5x + 1)(3x^2 - x - 4)\), we will use the distributive property (also known as the FOIL method for binomials, but here it applies to trinomials as well).

We need to multiply each term in the first polynomial by each term in the second polynomial.

### Step 1: Distribute \(2x^2\)

1. \(2x^2 \cdot 3x^2 = 6x^4\)
2. \(2x^2 \cdot (-x) = -2x^3\)
3. \(2x^2 \cdot (-4) = -8x^2\)

### Step 2: Distribute \(-5x\)

1. \(-5x \cdot 3x^2 = -15x^3\)
2. \(-5x \cdot (-x) = 5x^2\)
3. \(-5x \cdot (-4) = 20x\)

### Step 3: Distribute \(1\)

1. \(1 \cdot 3x^2 = 3x^2\)
2. \(1 \cdot (-x) = -x\)
3. \(1 \cdot (-4) = -4\)

### Step 4: Combine all the results

Now, let's combine all the terms together:

\[
6x^4 + (-2x^3 - 15x^3) + (-8x^2 + 5x^2 + 3x^2) + (20x - x) - 4
\]

### Step 5: Combine like terms

1. For \(x^4\): \(6x^4\)
2. For \(x^3\): \(-2x^3 - 15x^3 = -17x^3\)
3. For \(x^2\): \(-8x^2 + 5x^2 + 3x^2 = 0\)
4. For \(x\): \(20x - x = 19x\)
5. For the constant: \(-4\)

### Final Result

Putting it all together, we get:

\[
6x^4 - 17x^3 + 19x - 4
\]

So the expanded expression in standard form is:

\[
\boxed{6x^4 - 17x^3 + 19x - 4}
\]