Help with sum and product of roots.

Question

I'm having trouble with a question from my textbook relating to roots of an equation. This is it:

Let a and b be the roots of the equation:

$x^2-x-5=0$

Find the value of $(a^2+4b-1)(b^2+4a-1)$, without calculating values of $a$ and $b$.

What I do know however, is that the book has hinted towards sum and product of roots in which I have determined that the SUM OF ROOTS is $1$ and the PRODUCT OF ROOTS is $-5$. So really I'm just having difficulty finding the value. I don't want a direct answer can I have a few hints to get myself closer to getting the answer?

Answer 1

HInt: You have $a+b=1$ and $ab=-5$. Now expand the product $(a^2+4b-1)(b^2+4a-1)=(ab)^2+4(a^3+b^3)+16ab-4(a+b)-(a^2+b^2)+1$, then try to express the terms $a^3+b^3$ and $a^2+b^2$ in terms of $ab$ and $a+b$.

Answer 2

Hint expand the expression and use $a^3+b^3=(a+b)^3-3ab(a+b),a^2+b^2=(a+b)^2-2ab$ and then it's all easy.

Answer 3

You know that $a+b=1$ and $ab=-5$. You also know that $a^2-a-5=0=b^2-b-5$ so you can replace $a^2$ by $a+5$ and $b^2$ by $b+5$ everywhere you see them.