Factor polynomial $$ 2x^{2}+63x+145 $$

$$ 2x^{2}+63x+145 = ( 2x+5 )( x+29 ) $$

Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 2 , b = 63 ~ \text{ and } ~ c = 145 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 145} $.

$$ a \cdot c = 290 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 290 $ and add to $ b = 63 $.

Step 4: All pairs of numbers with a product of $ 290 $ are:

PRODUCT = 290
1     290	-1     -290
2     145	-2     -145
5     58	-5     -58
10     29	-10     -29

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 63 }$

PRODUCT = 290 and SUM = 63
1     290	-1     -290
2     145	-2     -145
5     58	-5     -58
10     29	-10     -29

Step 6: Replace middle term $ 63 x $ with $ 58x+5x $:

$$ 2x^{2}+63x+145 = 2x^{2}+58x+5x+145 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 5 $ out of the last two terms.

$$ 2x^{2}+58x+5x+145 = 2x\left(x+29\right) + 5\left(x+29\right) = \left(2x+5\right) \left(x+29\right) $$

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