$\text{a)}$  

$\{\begin{array}{l}F\left(1\right)=0\\ F\left(-2\right)=21\end{array}$  

$\{\begin{array}{l}2\cdot 1^2+m\cdot 1+n=0\\ 2\cdot {\left(-2\right)}^2+m\cdot \left(-2\right)+n=21\end{array}$  

$\{\begin{array}{l}2+m+n=0\\ 2\cdot 4-2m+n=21\end{array}$  

$\{\begin{array}{l}m+n=-2\\ 8-2m+n=21\end{array}$  

$\{\begin{array}{l}m+n=-2\\ n=2m+13\end{array}$  

Podstawiamy n=2m+13 do pierwszego równania w układzie.

$\{\begin{array}{l}m+2m+13=-2\\ n=2m+13\end{array}$  

$\{\begin{array}{l}3m=-15\mid :3\\ n=2m+13\end{array}$  

$\{\begin{array}{l}m=-5\\ n=2m+13\end{array}$  

Podstawiamy m=-5 do drugiego równania w układzie.

$\{\begin{array}{l}m=-5\\ n=2\cdot \left(-5\right)+13\end{array}$  

$\{\begin{array}{l}m=-5\\ n=-10+13\end{array}$  

$\{\begin{array}{l}m=-5\\ n=3\end{array}$  

---

$\text{b)}$  

$\{\begin{array}{l}F\left(-1\right)=1\\ F\left(2\right)=13\end{array}$  

$\{\begin{array}{l}{\left(-1\right)}^3+m\cdot {\left(-1\right)}^2+n\cdot \left(-1\right)-5=1\\ 2^3+m\cdot 2^2+n\cdot 2-5=13\end{array}$  

$\{\begin{array}{l}-1+m-n-5=1\\ 8+4m+2n-5=13\end{array}$  

$\{\begin{array}{l}m-n=7\\ 4m+2n=10\mid :2\end{array}$  

$\{\begin{array}{l}m-n=7\\ 2m+n=5\end{array}$  

Dodajemy równania stronami.

$3m=12\mid :3$  

$m=4$  

Podstawiamy m=4 do dowolnego równania w układzie.

$\{\begin{array}{l}m=4\\ 2m+n=5\end{array}$  

$\{\begin{array}{l}m=4\\ 2\cdot 4+n=5\end{array}$  

$\{\begin{array}{l}m=4\\ 8+n=5\end{array}$  

$\{\begin{array}{l}m=4\\ n=-3\end{array}$  

---

$\text{c)}$  

$\{\begin{array}{l}F\left(-2\right)=-1\\ F\left(0\right)=3\end{array}$  

$\{\begin{array}{l}-{\left(-2\right)}^4-2\cdot {\left(-2\right)}^3+m\cdot \left(-2\right)+n=-1\\ 0-2\cdot 0+m\cdot 0+n=3\end{array}$  

$\{\begin{array}{l}-16-2\cdot \left(-8\right)-2m+n=-1\\ n=3\end{array}$  

$\{\begin{array}{l}-16+16-2m+n=-1\\ n=3\end{array}$  

$\{\begin{array}{l}-2m+n=-1\\ n=3\end{array}$  

Podstawiamy n=3 do pierwszego równania w układzie.

$\{\begin{array}{l}-2m+3=-1\\ n=3\end{array}$  

$\{\begin{array}{l}-2m=-4\mid :\left(-2\right)\\ n=3\end{array}$  

$\{\begin{array}{l}m=2\\ n=3\end{array}$  

---

$\text{d)}$  

$\{\begin{array}{l}F\left(-1\right)=2\\ F\left(2\right)=-31\end{array}$  

$\{\begin{array}{l}-5\cdot {\left(-1\right)}^3+m\cdot \left(-1\right)+n=2\\ -5\cdot 2^3+m\cdot 2+n=-31\end{array}$  

$\{\begin{array}{l}-5\cdot \left(-1\right)-m+n=2\\ -5\cdot 8+2m+n=-31\end{array}$  

$\{\begin{array}{l}5-m+n=2\\ -40+2m+n=-31\end{array}$  

$\{\begin{array}{l}-m+n=-3\\ 2m+n=9\end{array}$  

$\{\begin{array}{l}-m+n=-3\\ n=9-2m\end{array}$  

Podstawiamy n=9-2m do pierwszego równania w układzie.

$\{\begin{array}{l}-m+9-2m=-3\\ n=9-2m\end{array}$  

$\{\begin{array}{l}-3m=-12\mid :\left(-3\right)\\ n=9-2m\end{array}$  

$\{\begin{array}{l}m=4\\ n=9-2m\end{array}$  

Podstawiamy m=4 do drugiego równania w układzie.

$\{\begin{array}{l}m=4\\ n=9-2\cdot 4\end{array}$  

$\{\begin{array}{l}m=4\\ n=9-8\end{array}$  

$\{\begin{array}{l}m=4\\ n=1\end{array}$  

---

$\text{e)}$  

$\{\begin{array}{l}F\left(1\right)=-5\\ F\left(-1\right)=-1\end{array}$  

$\{\begin{array}{l}2\cdot 1^4+m\cdot 1^3+1+n=-5\\ 2\cdot {\left(-1\right)}^4+m\cdot {\left(-1\right)}^3+\left(-1\right)+n=-1\end{array}$  

$\{\begin{array}{l}2+m+1+n=-5\\ 2-m-1+n=-1\end{array}$  

$\{\begin{array}{l}m+n=-8\\ -m+n=-2\end{array}$  

Dodajemy równania stronami.

$2n=-10\mid :2$  

$n=-5$  

Podstawiamy n=-5 do dowolnego równania w układzie.

$\{\begin{array}{l}m+n=-8\\ n=-5\end{array}$  

$\{\begin{array}{l}m+\left(-5\right)=-8\\ n=-5\end{array}$  

$\{\begin{array}{l}m=-3\\ n=-5\end{array}$  

---

$\text{f)}$  

$\{\begin{array}{l}F\left(2\sqrt{2}\right)=76\\ F\left(-\sqrt{2}\right)=4\end{array}$  

$\{\begin{array}{l}m\cdot {\left(2\sqrt{2}\right)}^4+n\cdot {\left(2\sqrt{2}\right)}^2-4=76\\ m\cdot {\left(-\sqrt{2}\right)}^4+n\cdot {\left(-\sqrt{2}\right)}^2-4=4\end{array}$  

$\{\begin{array}{l}64m+8n=80\mid :8\\ 4m+2n=8\mid :2\end{array}$  

$\{\begin{array}{l}8m+n=10\\ 2m+n=4\end{array}$  

$\{\begin{array}{l}8m+n=10\\ n=4-2m\end{array}$  

Podstawiamy n=4-2m do pierwszego równania w układzie.

$\{\begin{array}{l}8m+4-2m=10\\ n=4-2m\end{array}$  

$\{\begin{array}{l}6m=6\mid :6\\ n=4-2m\end{array}$  

$\{\begin{array}{l}m=1\\ n=4-2m\end{array}$  

Podstawiamy m=1 do drugiego równania w układzie.

$\{\begin{array}{l}m=1\\ n=4-2\cdot 1\end{array}$  

$\{\begin{array}{l}m=1\\ n=4-2\end{array}$  

$\{\begin{array}{l}m=1\\ n=2\end{array}$  

---

$\text{g)}$  

$\{\begin{array}{l}F\left(\sqrt{2}\right)=5\\ F\left(\sqrt{3}\right)=8\end{array}$  

$\{\begin{array}{l}m\cdot {\left(\sqrt{2}\right)}^4+n\cdot {\left(\sqrt{2}\right)}^2+5=5\\ m\cdot {\left(\sqrt{3}\right)}^4+n\cdot {\left(\sqrt{3}\right)}^2+5=8\end{array}$  

$\{\begin{array}{l}4m+2n=0\mid :2\\ 9m+3n=3\mid :3\end{array}$  

$\{\begin{array}{l}2m+n=0\\ 3m+n=1\end{array}$  

$\{\begin{array}{l}2m+n=0\\ n=1-3m\end{array}$  

Podstawiamy n=1-3m do pierwszego równania w układzie.

$\{\begin{array}{l}2m+1-3m=0\\ n=1-3m\end{array}$  

$\{\begin{array}{l}-m+1=0\\ n=1-3m\end{array}$  

$\{\begin{array}{l}1=m\\ n=1-3m\end{array}$  

Podstawiamy m=1 do drugiego równania w układzie.

$\{\begin{array}{l}m=1\\ n=1-3\cdot 1\end{array}$  

$\{\begin{array}{l}m=1\\ n=1-3\end{array}$  

$\{\begin{array}{l}m=1\\ n=-2\end{array}$