Question

Expand the expression to a polynomial in standard form:
(4x+5)(x^2 −2x+5)

Answer 1

To expand the expression \((4x + 5)(x^2 - 2x + 5)\), we will use the distributive property (also known as the FOIL method for binomials). Each term in the first polynomial will be multiplied by each term in the second polynomial.

1. Distributing \(4x\):
\[
4x \cdot x^2 = 4x^3
\]
\[
4x \cdot (-2x) = -8x^2
\]
\[
4x \cdot 5 = 20x
\]

2. Distributing \(5\):
\[
5 \cdot x^2 = 5x^2
\]
\[
5 \cdot (-2x) = -10x
\]
\[
5 \cdot 5 = 25
\]

Now, we can combine all these results:
\[
4x^3 - 8x^2 + 20x + 5x^2 - 10x + 25
\]

Next, we combine the like terms:
- The \(x^2\) terms: \(-8x^2 + 5x^2 = -3x^2\)
- The \(x\) terms: \(20x - 10x = 10x\)

Putting it all together, the expanded polynomial in standard form is:
\[
4x^3 - 3x^2 + 10x + 25
\]

Thus, the final answer is:
\[
\boxed{4x^3 - 3x^2 + 10x + 25}
\]