Answer:
[tex]a+2b-d=1[/tex]
Step-by-step explanation:
We are given that:
[tex](ax^2+bx+3)(x+d)=x^3+6x^2+11x+12[/tex]
And we want to determine:
[tex]a+2b-d[/tex]
So, we will determine our unknowns first.
We can distribute our expression:
[tex]=(ax^2+bx+3)x+(ax^2+bx+3)d[/tex]
Distribute:
[tex]=ax^3+bx^2+3x+adx^2+bdx+3d[/tex]
Rearranging gives:
[tex]=(ax^3)+(bx^2+adx^2)+(bdx+3x)+3d[/tex]
Factoring out the variable yields:
[tex]=(a)x^3+(b+ad)x^2+(bd+3)x+d(3)[/tex]
Since we know that our expression equals:
[tex]x^3+6x^2+11x+12[/tex]
This means that each of the unknown terms in front of each variable corresponds with the coefficient of the resulting equation. Therefore:
[tex]\begin{aligned} a&=1\\ b+ad&=6\\bd+3&=11\\3d&=12\end{aligned}[/tex]
Solving the first and fourth equation yields that:
[tex]a=1\text{ and } d=4[/tex]
Then the second and third equations become:
[tex]b+(1)(4)=6\text{ and } b(4)+3=11[/tex]
And solving for b now yields that:
[tex]b=2\stackrel{\checkmark}{=}2[/tex]
Therefore, we know that:
[tex]a=1, b=2\text{ and } d=4[/tex]
For the equation:
[tex](x^2+2x+3)(x+4)=x^3+6x^2+11x+ 12[/tex]
Then the expression:
[tex]a+2b-d[/tex]
Can be evaluated as:
[tex]=(1)+2(2)-4[/tex]
Evaluate:
[tex]=1+4-4=1[/tex]
Hence, our final answer is 1.
