How to solve the equation $(2x + 1)(2x + 3) = 143$ without using the Quadratic Formula? I have been a bit stuck on this question.
The product of two consecutive odd numbers is $143$. Find the next numbers.
I have made this into:
$$
(2x+1)(2x+3)=143.
$$
I got $x_1 = -7$ and $x_2 = -5$ for this, which doesn't seem right.
How can I fix this? (By the way we are only able to solve for $x$ by using factorising and not with the quadratic formula).

An idea: decompose in primes product $\;143=11\cdot13\;$ . 
Observe that we're given $\;(2x+1)(2x+3)=143\;$ , and the left side is the product of two consecutive odd integers. Once you get the prime product for $\;143\;$ you already have not many choices: one must be $\;11\;$ , and the other one $\;13\;$ ...or both negative: $\;-11,\,-13\;$ , and you get:
$$\begin{cases}2x+1=11\\2x+3=13\end{cases}\implies \color{red}{x=5}\;,\;\;\begin{cases}2x+1=-13\\2x+3=-11\end{cases}\implies \color{red}{x=-7}$$
