Verified answer

[tex]\displaystyle\bf\\x=\sqrt{7} +1\\\\\\x^{2} +4x\sqrt{7} +5=(\sqrt{7} +1)^{2} +4\sqrt{7} \cdot(\sqrt{7} +1)+5=\\\\\\=(\sqrt{7} )^{2} +2\cdot \sqrt{7} \cdot 1+1^{2} +4\sqrt{7} \cdot \sqrt{7} +4\sqrt{7} \cdot 1+5=\\\\\\=7+2\sqrt{7} +1+28+4\sqrt{7} +5=(7+1+28+5)+(2\sqrt{7} +4\sqrt{7} )=\\\\\\=\boxed{41+6\sqrt{7} }[/tex]