How do you factor \[2{t^2} + 7t + 3\] ?

Hint: Here in this question, we have to find the factors, the given equation is in the form of a quadratic equation. This is a quadratic equation for the variable t. By using the formula \[t = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] , we can determine the roots of the equation.

Complete step-by-step answer:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factorising or by using the formula \[t = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . So the equation is written as \[2{t^2} + 7t + 3\] .
In general, the quadratic equation is represented as \[a{x^2} + bx + c = 0\] , when we compare the above equation to the general form of equation the values are as follows. a=2 b=7 and c=3. Now substituting these values to the formula for obtaining the roots we have
 \[roots = \dfrac{{ - 7 \pm \sqrt {{7^2} - 4(2)(3)} }}{{2(2)}}\]
On simplifying the terms, we have
 \[ \Rightarrow roots = \dfrac{{ - 7 \pm \sqrt {49 - 24} }}{4}\]
Now subtract 24 from 49 we get
 \[ \Rightarrow roots = \dfrac{{ - 7 \pm \sqrt {25} }}{4}\]
The number 25 is a perfect square so we can take out from square root we have
 \[ \Rightarrow roots = \dfrac{{ - 7 \pm 5}}{4}\]
Therefore, we have \[root1 = \dfrac{{ - 7 + 5}}{4}\] or \[root2 = \dfrac{{ - 7 - 5}}{4}\] .
On simplifying we get
 \[root_1 = \dfrac{{ - 2}}{4}\] or \[root_2 = \dfrac{{ - 12}}{4}\] .
On further simplification we have
 \[root_1 = \dfrac{{ - 1}}{2}\] or \[root_2 = - 3\] .
Substituting the roots values we have
 \[ \Rightarrow \left( {t - \left( {\dfrac{{ - 1}}{2}} \right)} \right)\left( {t - \left( { - 3} \right)} \right)\]
On simplifying we have
 \[ \Rightarrow \left( {t + \dfrac{1}{2}} \right)\left( {t + 3} \right)\]
Hence we have found the factors for the given equation
So, the correct answer is “$\left( {t + \dfrac{1}{2}} \right)\left( {t + 3} \right)$”.

Note: he quadratic equation can be solved by using the factorisation method and we also find the roots by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.