Factorise $10 p^{2} + 11 p + 3$ and choose the option representing the correct factored form.

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(5p - 3) (2p - 1)

(2p + 3) (5p + 1)

(5p + 3) (2p + 1)

(2p + 3) (5p + 5)

answer is A.

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Detailed Solution

The given quadratic equation $10 p^{2} + 11 p + 3$ can be written in the standard form as follows,
$10 p^{2} + 11 p + 3$ .
Now, it is of the form $a p^{2} + b p + c$ , where a=10, b=11 and c=3. In order to factorise these types of equations, we need to find two factors, say x and y such that the following two conditions are satisfied,
$a c = x y$
$x + y = b$
According to the above conditions, the product of first and last constant term of the expression will be,
$10 \times 3 = 30$ .
Next, we will have to find the factors of the product, i.e., 30, such that their sum or difference is equal to the middle term.
So, we have the factors as $5 \times 6 = 30$ , which is the product of the first and last terms of the equation, while their sum, i.e., $5 + 6 = 11$ is the middle term of the given equation.
Now, we will factorise the expression using the factors and splitting the middle term as follows,
$\begin{matrix} 10 p^{2} + 11 p + 3 = 10 p^{2} + 5 p + 6 p + 3 \\ = 5 p (2 p + 1) + 3 (2 p + 1) \\ = (2 p + 1) (5 p + 3) \end{matrix}$
So, the factored form of $10 p^{2} + 11 p + 3$ is $(2 p + 1) (5 p + 3)$ .