Question

multiply and write in standard form.
1
$(x - 1)(x^2 + 2x + 4)$
2
$(x + 2)(x^2 + 5x - 10)$
3
$(3 - x)(2x^2 + 3x - 15)$

Explanation:

Problem 1: \((x - 1)(x^2 + 2x + 4)\)

Step 1: Distribute \(x\) and \(-1\)

Using the distributive property (FOIL for polynomials), we multiply \(x\) by each term in \((x^2 + 2x + 4)\) and \(-1\) by each term in \((x^2 + 2x + 4)\):
\(x(x^2 + 2x + 4) - 1(x^2 + 2x + 4)\)
\(= x^3 + 2x^2 + 4x - x^2 - 2x - 4\)

Step 2: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms:
\(x^3 + (2x^2 - x^2) + (4x - 2x) - 4\)
\(= x^3 + x^2 + 2x - 4\)

Step 1: Distribute \(x\) and \(2\)

Multiply \(x\) by each term in \((x^2 + 5x - 10)\) and \(2\) by each term in \((x^2 + 5x - 10)\):
\(x(x^2 + 5x - 10) + 2(x^2 + 5x - 10)\)
\(= x^3 + 5x^2 - 10x + 2x^2 + 10x - 20\)

Step 2: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms (note that \(-10x + 10x = 0\)):
\(x^3 + (5x^2 + 2x^2) + (-10x + 10x) - 20\)
\(= x^3 + 7x^2 - 20\)

Step 1: Distribute \(3\) and \(-x\)

Multiply \(3\) by each term in \((2x^2 + 3x - 15)\) and \(-x\) by each term in \((2x^2 + 3x - 15)\):
\(3(2x^2 + 3x - 15) - x(2x^2 + 3x - 15)\)
\(= 6x^2 + 9x - 45 - 2x^3 - 3x^2 + 15x\)

Step 2: Combine like terms

Rearrange the terms in descending order of exponents and combine like terms:
\(-2x^3 + (6x^2 - 3x^2) + (9x + 15x) - 45\)
\(= -2x^3 + 3x^2 + 24x - 45\)

Answer:

\(x^3 + x^2 + 2x - 4\)

Problem 2: \((x + 2)(x^2 + 5x - 10)\)