Expand the expression to a polynomial in standard form:

Question

left parenthesis, 4, x, squared, minus, x, plus, 9, right parenthesis, left parenthesis, x, squared, plus, 2, x, plus, 1, right parenthesis
(4x
2
−x+9)(x
2
+2x+1)

Solve the problem just like the example below
⬇️
Expand the expression to a polynomial in standard form:
left parenthesis, x, squared, minus, x, minus, 2, right parenthesis, left parenthesis, minus, x, squared, plus, 3, x, minus, 6, right parenthesis
(x
2
−x−2)(−x
2
+3x−6)

minus, x, squared
−x
2
plus, 3, x
+3x
minus, 6
−6
x, squared
x
2
minus, x, to the power 4
−x
4
3, x, cubed
3x
3
minus, 6, x, squared
−6x
2
minus, x
−x
x, cubed
x
3
minus, 3, x, squared
−3x
2
6, x
6x
minus, 2
−2
2, x, squared
2x
2
minus, 6, x
−6x
12
12
Combine like terms:
Combine like terms:
minus, x, to the power 4 , plus, 4, x, cubed, minus, 7, x, squared, plus, 12
−x
4
+4x
3
−7x
2
+12

Answer 1

To expand the expression \((4x^2 - x + 9)(x^2 + 2x + 1)\), we will follow a similar process as in the example.

Step 1: Distribute each term in the first polynomial to each term in the second polynomial.

1. Distributing \(4x^2\):

   • \(4x^2 \cdot x^2 = 4x^4\)
   • \(4x^2 \cdot 2x = 8x^3\)
   • \(4x^2 \cdot 1 = 4x^2\)

2. Distributing \(-x\):

   • \(-x \cdot x^2 = -x^3\)
   • \(-x \cdot 2x = -2x^2\)
   • \(-x \cdot 1 = -x\)

3. Distributing \(9\):

   • \(9 \cdot x^2 = 9x^2\)
   • \(9 \cdot 2x = 18x\)
   • \(9 \cdot 1 = 9\)

Step 2: Combine all these results together: \[ 4x^4 + 8x^3 + 4x^2 - x^3 - 2x^2 - x + 9x^2 + 18x + 9 \]

Step 3: Gather all like terms:

• \(4x^4\) (only one term)
• The \(x^3\) terms: \(8x^3 - x^3 = 7x^3\)
• The \(x^2\) terms: \(4x^2 - 2x^2 + 9x^2 = 11x^2\)
• The \(x\) terms: \(-x + 18x = 17x\)
• The constant term: \(9\) (only one term)

Final Step: Write the polynomial in standard form: \[ 4x^4 + 7x^3 + 11x^2 + 17x + 9 \]

Thus, the expanded form of \((4x^2 - x + 9)(x^2 + 2x + 1)\) in standard form is: \[ \boxed{4x^4 + 7x^3 + 11x^2 + 17x + 9} \]