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$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

First, we are going to assume that the breadth of the enclosure is $x$. So, ${\text{breadth = }}x$

It is also given that the length is 4m more than thrice its width.

Hence we have, ${\text{length = 3}}x{\text{ + 4}}$

Now we have the length and breadth of the enclosure that is rectangle.

It is also given that the area of the enclosure is $480{\text{ }}{{\text{m}}^2}$

Now we are going to consider the formula of area of the enclosure, that is, rectangle.

Area of enclosure = ${length \times breadth}$ --(ii)

Now we are going to substitute the values on the above equation. Then we get,

$x(3x + 4) = 480$

Now we are going to simplify the above equation.

$3{x^2} + 4x = 480$

Now we are going to subtract 480 on both sides so that the right hand side will be zero.

$3{x^2} + 4x - 480 = 0$ ----(iii)

We already know that any equation of the form $a{x^2} + bx + c = 0$ is called a quadratic equation.

So, $3{x^2} + 4x - 480 = 0$ is a quadratic equation where $a = 3$, $b = 4$ and $c = - 480$

Now we are going to solve the above quadratic equation to get the value of the $x$

To solve for $x$, we are going to use the quadratic formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Now we are going to substitute the values of a, b and c on the above formula. Then we get,

$x = \dfrac{{ - 4 \pm \sqrt {{4^2} - 4 \times 3 \times ( - 430)} }}{{2 \times 3}}$

We are going to simplify the above equation.

$x = \dfrac{{ - 4 \pm \sqrt {16 + 12 \times 480} }}{6}$

Now we are going to use the BODMAS rule to simplify the term that is in the square root. First multiplication holds and then the addition holds.

$x = \dfrac{{ - 4 \pm \sqrt {16 + (12 \times 480)} }}{6}$

First we multiply, $12 \times 480$,

$x = \dfrac{{ - 4 \pm \sqrt {16 + 5760} }}{6}$

Then we add, 16+5760,

$x = \dfrac{{ - 4 \pm \sqrt {5776} }}{6}$

Now we are going to substitute $\sqrt {5776} = 76$ on the above equation. Then we get,

$x = \dfrac{{ - 4 \pm 76}}{6}$

Length and breadth are never taken as a negative value. So, we can skip the negative operation of the above equation and we are going to consider only addition operations.

\[x = \dfrac{{ - 4 + 76}}{6}\]

\[x = \dfrac{{ - 4 + 76}}{6}\]

\[x = \dfrac{{72}}{6}\]

We are going to simplify the above equation to get the value of \[x\]

Therefore, \[x = 12\]

Hence the breadth of the enclosure, \[x = 12{\text{ m}}\]

Now we are going to find the length of the enclosure.

It is already given that, ${\text{length = 3}}x{\text{ + 4}}$

We are going to substitute \[x = 12{\text{ m}}\]

Length of the closure = $3x + 4$ = $3 \times 12 + 4$

Now we are going to use the BODMAS rule to solve the above equation.

Hence, ${\text{length = }}36 + 4 = 40{\text{ m}}$

So, the length of the enclosure is ${\text{40 m}}$ and the breadth of the enclosure is \[{\text{12 m}}\].