To solve the polynomial multiplication problem given as:
[tex]\[ 6x^2 \left( x^4 - 2x^3 + 5x^2 + x - 1 \right) \][/tex]
we need to distribute [tex]\( 6x^2 \)[/tex] across each term inside the parentheses. We'll proceed by multiplying [tex]\( 6x^2 \)[/tex] with each term individually and then combining all the terms together.
Here is the step-by-step solution:
1. Multiply [tex]\( 6x^2 \)[/tex] by the first term [tex]\( x^4 \)[/tex]:
[tex]\[ 6x^2 \cdot x^4 = 6x^{2+4} = 6x^6 \][/tex]
2. Multiply [tex]\( 6x^2 \)[/tex] by the second term [tex]\( -2x^3 \)[/tex]:
[tex]\[ 6x^2 \cdot (-2x^3) = 6 \cdot -2 \cdot x^{2+3} = -12x^5 \][/tex]
3. Multiply [tex]\( 6x^2 \)[/tex] by the third term [tex]\( 5x^2 \)[/tex]:
[tex]\[ 6x^2 \cdot 5x^2 = 6 \cdot 5 \cdot x^{2+2} = 30x^4 \][/tex]
4. Multiply [tex]\( 6x^2 \)[/tex] by the fourth term [tex]\( x \)[/tex]:
[tex]\[ 6x^2 \cdot x = 6 \cdot x^{2+1} = 6x^3 \][/tex]
5. Multiply [tex]\( 6x^2 \)[/tex] by the fifth term [tex]\( -1 \)[/tex]:
[tex]\[ 6x^2 \cdot (-1) = 6 \cdot -1 \cdot x^2 = -6x^2 \][/tex]
Now, combine all the distributed terms together:
[tex]\[ 6x^6 - 12x^5 + 30x^4 + 6x^3 - 6x^2 \][/tex]
Thus, the expanded and simplified form of the polynomial [tex]\( 6x^2 \left( x^4 - 2x^3 + 5x^2 + x - 1 \right) \)[/tex] is:
[tex]\[ 6x^6 - 12x^5 + 30x^4 + 6x^3 - 6x^2 \][/tex]
This is the final result.
