\[\boxed{\text{1015.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[\textbf{а)}\ (x + 1)^{2} \geq 4x\]
\[x^{2} + 2x + 1 - 4x \geq 0\]
\[x^{2} - 2x + 1 \geq 0\]
\[(x - 1)^{2} \geq 0 \Longrightarrow верно\ при\ \]
\[любом\ x:\ \ ч.т.д.\]
\[\textbf{б)}\ (3b + 1)^{2} > 6b\]
\[9b^{2} + 6b + 1 - 6b > 0\]
\[9b^{2} > - 1 \Longrightarrow верно\ при\ любом\]
\[\ b,\ так\ как\ 9b^{2} \geq 0:\ \ ч.т.д.\]
\[\textbf{в)}\ 4 \cdot (x + 2) < (x + 3)^{2} - 2x\]
\[4x + 8 < x^{2} + 6x + 9 - 2x\]
\[x^{2} + 1 > 0 \Longrightarrow верно\ при\ \]
\[любом\ x,\ так\ как\ x^{2} \geq 0:\ \ ч.т.д.\]
\[\textbf{г)}\ 1 + (m + 2)^{2} > 3 \cdot (2m - 1)\]
\[1 + m^{2} + 4m + 4 > 6m - 3\]
\[m^{2} - 2m + 8 > 0\]
\[\left( m^{2} - 2m + 1 \right) + 7 > 0\]
\[(m - 1)^{2} + 7 > 0 \Longrightarrow верно\ при\ \]
\[любом\ m,\ так\ как\]
\[\ (m - 1)^{2} \geq 0:ч.т.д.\]