Given the problem:

• Original length of the rectangular garden $l = 7$ meters
• Original width of the rectangular garden $w = 4$ meters
• Length increased by $x$ meters, making the new length $l' = 7 + x$ meters
• Width increased by $3x$ meters, making the new width $w' = 4 + 3x$ meters
• New area of the garden $A = 90$ square meters

We need to form and solve the quadratic equation based on this information.

Step 1: Express the new area in terms of $x$.

The new area of the rectangular garden is given by:

$A = l' \times w'$

Substituting the values:

$90 = (7 + x)(4 + 3x)$

Step 2: Expand the equation.

Expand the right side of the equation:

$90 = 7 \times 4 + 7 \times 3x + x \times 4 + x \times 3x$

$90 = 28 + 21x + 4x + 3x^2$

Combine like terms:

$90 = 3x^2 + 25x + 28$

Step 3: Rearrange the equation into standard quadratic form.

Move all terms to one side of the equation:

$3x^2 + 25x + 28 - 90 = 0$

$3x^2 + 25x - 62 = 0$

This is the required quadratic equation:

$\boxed{3x^2 + 25x - 62 = 0}$

Would you like to see the steps for solving this quadratic equation or need any further clarifications?

Here are some related questions to explore:

1. How do you solve a quadratic equation using the quadratic formula?
2. What are the possible values of $x$ for the given quadratic equation?
3. How does changing the length and width affect the area of a rectangle?
4. Can you find the roots of the quadratic equation $3x^2 + 25x - 62 = 0$?
5. What are the applications of quadratic equations in real-life problems like this one?

Tip: When forming quadratic equations from word problems, always identify the variables and express the known relationships in terms of these variables.