(2n+7)>(n+3)² prove by mathematical induction

Let P (n) =(2n + 7) < (n + 3)²
For n = 1
{tex}P(1) = (2 \times 1 + 7) < {(1 + 3)^2} \Rightarrow 9 < 16{/tex}
{tex}\therefore {/tex} P ( 1) is true
Let P(n) be true for n = k
{tex}\therefore P(k) = (2k + 7) < {(k + 3)^2}{/tex} ....(1)
For n = k + 1 
P (k + 1) = 2 (k + 1) + 7 < (k + 1 + 3)²
{tex} \Rightarrow {/tex} 2(k + 1) + 7 < (k + 4)²
From (1)
2k + 7 < (k + 3)²
Adding 2 on both sides
2k + 7 + 2 < (k + 3)² + 2 {tex} \Rightarrow {/tex} 2(k + 1) + 7 < k² + 9 + 6k + 2
{tex} \Rightarrow {/tex} 2(k + 1) + 7 < k² +6k + 11 < k² + 8k + 16
2(k + 1) + 7 <(k + 4)²
{tex}\therefore {/tex} P (k + 1) is true
Thus P(k) is true {tex} \Rightarrow {/tex} P(k + 1) is true
Hence by principle of mathematical induction, P (n) is true for all {tex}n \in N{/tex}.