Question

Expand the expression to a polynomial in the standard form: (2x+5)(2x^2+x-3)

Answer 1

To expand the expression \((2x+5)(2x^2+x-3)\), we will use the distributive property (also known as the FOIL method for binomials, but in this case, we have a binomial multiplied by a trinomial).

Let's distribute \(2x\) and \(5\) across the trinomial \(2x^2+x-3\):

1. **Distributing \(2x\):**
\[
2x \cdot 2x^2 = 4x^3
\]
\[
2x \cdot x = 2x^2
\]
\[
2x \cdot (-3) = -6x
\]

2. **Distributing \(5\):**
\[
5 \cdot 2x^2 = 10x^2
\]
\[
5 \cdot x = 5x
\]
\[
5 \cdot (-3) = -15
\]

Now, we combine all the results:

\[
4x^3 + 2x^2 - 6x + 10x^2 + 5x - 15
\]

Next, we combine like terms:

- The \(x^2\) terms: \(2x^2 + 10x^2 = 12x^2\)
- The \(x\) terms: \(-6x + 5x = -1x\) or \(-x\)

Putting all together, we have:

\[
4x^3 + 12x^2 - x - 15
\]

So, the final expanded expression in standard form is:

\[
\boxed{4x^3 + 12x^2 - x - 15}
\]