In a class test, the sum of the marks obtained by P in mathematics and science is 28. Had he got 3 more marks in mathematics and 4 marks less in science, the product of marks obtained in the two subjects would have been 180. Find the marks obtained by him in the two subjects separately.

Explanation

To solve the problem step by step, we will define the variables, set up equations based on the information given, and then solve for the marks obtained by P in mathematics and science.

### Step 1: Define the Variables
Let:
- \( x \) = marks obtained in Mathematics
- \( y \) = marks obtained in Science

### Step 2: Set Up the Equations
From the problem, we know two things:
1. The sum of the marks in Mathematics and Science is 28:
\[ x + y = 28 \quad \text{(Equation 1)} \]
2. If P had scored 3 more marks in Mathematics and 4 marks less in Science, the product of the marks would have been 180:
\[ (x + 3)(y - 4) = 180 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( y \) from Equation 1 into Equation 2
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 28 - x \]
Now substitute this into Equation 2:
\[ (x + 3)((28 - x) - 4) = 180 \]
This simplifies to:
\[ (x + 3)(24 - x) = 180 \]

### Step 4: Expand the Equation
Expanding the left side:
\[ x \cdot 24 - x^2 + 3 \cdot 24 - 3x = 180 \]
This simplifies to:
\[ 24x - x^2 + 72 - 3x = 180 \]
Combining like terms gives:
\[ 21x - x^2 + 72 = 180 \]

### Step 5: Rearrange the Equation
Rearranging the equation to set it to zero:
\[ -x^2 + 21x + 72 - 180 = 0 \]
This simplifies to:
\[ -x^2 + 21x - 108 = 0 \]
Multiplying through by -1 to make the leading coefficient positive:
\[ x^2 - 21x + 108 = 0 \quad \text{(Equation 3)} \]

### Step 6: Factor the Quadratic Equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to 108 and add to -21. The numbers are -12 and -9:
\[ (x - 12)(x - 9) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 12 = 0 \) → \( x = 12 \)
2. \( x - 9 = 0 \) → \( x = 9 \)

### Step 8: Find Corresponding \( y \) Values
Using Equation 1 to find \( y \):
1. If \( x = 12 \):
\[ y = 28 - 12 = 16 \]
2. If \( x = 9 \):
\[ y = 28 - 9 = 19 \]

### Conclusion
The marks obtained by P in Mathematics and Science are:
1. \( (12, 16) \) or
2. \( (9, 19) \)